linalg.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Primer of Linear Algebra and Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Notation\n",
    "\n",
    "${\\mathcal M}(m,n)$ will denote the set of $m$ by $n$ matrices.\n",
    "$A \\in {\\mathcal M}(m,n)$ has elements $A_{ij}$, $i \\in \\{1, \\ldots, m\\}$, $j \\in \\{1, \\ldots, n\\}$.\n",
    "\n",
    "We will identify ${\\mathcal M}(1,n)$ and ${\\mathcal M}(n,1)$ with $\\Re^n$, in a deliberate abuse\n",
    "of notation.\n",
    "Whether $x \\in \\Re^n$ should be considered a _column vector_ (an element of  ${\\mathcal M}(n,1)$)\n",
    "or a _row vector_ (an element of ${\\mathcal M}(1,n)$) should be clear from context or will be\n",
    "called out explicitly.\n",
    "For $a \\in {\\mathcal M}(1,n)$ or ${\\mathcal M}(n,1)$, we generally write only one index,\n",
    "suppressing the index that is always equal to 1\n",
    "(for instance, we would write $a_i$ in place of $a_{1i}$ or $a_{i1}$, respectively).\n",
    "\n",
    "It is convenient to write matrices as rectangular arrays, e.g.,\n",
    "\n",
    "\\begin{equation*} \n",
    "A = (A_{ij}) = \\begin{pmatrix} A_{11} & A_{12} & \\cdots & A_{1n} \\\\\n",
    "A_{21} & A_{22} & \\cdots & A_{2n} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "A_{m1} & A_{m2} & \\cdots & A_{mn} \n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "The $i$th _row_ of the matrix $A \\in {\\mathcal M}(m,n)$ is the matrix $B \\in {\\mathcal M}(1,n)$\n",
    "with elements $B_j = A_{ij}$, $j=1, \\ldots, n$.\n",
    "\n",
    "The $j$th _column_ of the matrix $A \\in {\\mathcal M}(m,n)$, is the matrix $C \\in {\\mathcal M}(m,j)$\n",
    "with elements $C_i = A_{ij}$, $i=1, \\ldots, m$.\n",
    "\n",
    "The _transpose_ of a matrix interchanges the row and column indices of the original matrix.\n",
    "If $A \\in {\\mathcal M}(m,n)$ then the transpose of $A$, denoted $A'$ or $A^T$, is an element of\n",
    "${\\mathcal M}(m,n)$ (If $A$ is an $m$ by $n$ matrix then $A'$ is an $n$ by $m$ matrix).\n",
    "If $A$ has elements $A_{ij}$, then\n",
    "$A' = A^T$ has elements $A_{ij}^T = A_{ji}$.\n",
    "\n",
    "__Scalar multiplication.__\n",
    "If $\\alpha \\in \\Re$ and $A \\in {\\mathcal M}(m,n)$, the matrix $\\alpha A \\in {\\mathcal M}(m,n)$ has elements $\\alpha A_{ij}$, $i \\in \\{1, \\ldots, m\\}$, $j \\in \\{1, \\ldots, n\\}$.\n",
    "I.e.,\n",
    "\n",
    "\\begin{equation*} \n",
    "\\alpha A = \\begin{pmatrix} \\alpha A_{11} & \\alpha A_{12} & \\cdots & \\alpha A_{1n} \\\\\n",
    "\\alpha A_{21} & \\alpha A_{22} & \\cdots & \\alpha A_{2n} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "\\alpha A_{m1} & \\alpha A_{m2} & \\cdots & \\alpha A_{mn} \n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "Let $-A = -1\\cdot A$ be the matrix with elements $(-A)_{ij} = -A_{ij}$:\n",
    "\n",
    "\\begin{equation*} \n",
    "-A = \\begin{pmatrix} -A_{11} & -A_{12} & \\cdots & -A_{1n} \\\\\n",
    "-A_{21} & -A_{22} & \\cdots & -A_{2n} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "-A_{m1} & -A_{m2} & \\cdots & -A_{mn} \n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "__Matrix addition.__\n",
    "If $A, B \\in {\\mathcal M}(m,n)$, then the matrix $A+B \\in {\\mathcal M}(m,n)$ has elements $A_{ij}+B_{ij}$, $i \\in \\{1, \\ldots, m\\}$, $j \\in \\{1, \\ldots, n\\}$:\n",
    "\n",
    "\\begin{equation*} \n",
    "A+B = \\begin{pmatrix} A_{11}+B_{11} & A_{12}+B_{12} & \\cdots & A_{1n}+B_{1n} \\\\\n",
    "A_{21}+B_{21} & A_{22}+B_{22} & \\cdots & A_{2n}+B_{2n} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "A_{m1}+B_{m1} & A_{m2}+B_{m2} & \\cdots & A_{mn}+B_{mn} \n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "Let $0_{mn} \\in {\\mathcal M}(m,n)$ denote the $m$ by $n$ matrix all of whose elements are zero; i.e., \n",
    "$(0_{mn})_{ij} = 0$, $i \\in \\{1, \\ldots, m\\}$, $j \\in \\{1, \\ldots, n\\}$:\n",
    "\n",
    "\\begin{equation*} \n",
    "0_{mn} = \\begin{pmatrix} 0 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 \\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "With these definitions, ${\\mathcal M}(m,n)$ is a linear vector space over the reals, with scalar identity element $1$ and vector identity element $0_{mn}$.\n",
    "\n",
    "__Transposition.__\n",
    "The _transpose_ of $A \\in {\\mathcal M}(m,n)$, $A^T = A' \\in {\\mathcal M}(n,m)$ has elements\n",
    "$(A')_{ij} = A_{ji}$, $i \\in \\{1, \\ldots, n\\}$, $j \\in \\{1, \\ldots, m\\}$:\n",
    "\n",
    "\\begin{equation*} \n",
    "A' = \\begin{pmatrix} A_{11} & A_{21} & \\cdots & A_{m1} \\\\\n",
    " A_{12} &  A_{22} & \\cdots &  A_{m2} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    " A_{1n} &  A_{2n} & \\cdots &  A_{mn} \n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "If $A = A'$, then $A$ is _symmetric_.\n",
    "\n",
    "Let ${\\mathbf 1}_{n} \\in {\\mathcal M}(n,n)$ denote the $n$ by $n$ _identity_ matrix \n",
    "with elements $(1_{n})_{ij} = 1$, $i=j \\in \\{1, \\ldots, n\\}$, and $(1_{n})_{ij} = 0$, $i \\ne j$:\n",
    "\n",
    "\\begin{equation*} \n",
    "1_{n} = \\begin{pmatrix} \n",
    "1 & 0 & \\cdots & 0 \\\\ \n",
    "0 & 1  & \\cdots & 0 \\\\  \n",
    "\\vdots & \\vdots  & \\ddots & 0\\\\\n",
    "0 & 0 &  \\cdots & 1 \n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "The identity matrix is an example of a _diagonal matrix_. A matrix $A$ is _diagonal_ if $A_{ij}=0$ \n",
    "whenever $i \\ne j$.\n",
    "\n",
    "__Matrix multiplication.__\n",
    "If $A \\in {\\mathcal M}(m,n)$ and $B \\in {\\mathcal M}(n,k)$, then $AB \\in {\\mathcal M}(m,k)$ is the matrix with\n",
    "elements $(AB)_{ij} = \\sum_{\\ell=1}^m A_{i \\ell}B_{\\ell j}$:\n",
    "\n",
    "\\begin{equation*} \n",
    "AB = \\begin{pmatrix} \n",
    "\\sum_{\\ell=1}^n A_{1 \\ell}B_{\\ell 1} & \\sum_{\\ell=1}^n A_{1 \\ell}B_{\\ell 2} & \\cdots & \n",
    "\\sum_{\\ell=1}^n A_{1 \\ell}B_{\\ell k} \\\\\n",
    "\\sum_{\\ell=1}^n A_{2 \\ell}B_{\\ell 1} & \\sum_{\\ell=1}^n A_{2 \\ell}B_{\\ell 2} & \\cdots & \n",
    "\\sum_{\\ell=1}^n A_{2 \\ell}B_{\\ell k} \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "\\sum_{\\ell=1}^n A_{m \\ell}B_{\\ell 1} & \\sum_{\\ell=1}^n A_{m \\ell}B_{\\ell 2} & \\cdots & \n",
    "\\sum_{\\ell=1}^n A_{m \\ell}B_{\\ell k}\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "If $A \\in {\\mathcal M}(m,n)$, then  $A {\\mathbf 1}_{n} = {\\mathbf 1}_{m} A = A$.\n",
    "\n",
    "A matrix $A$ is _square_ if it has the same number of rows and columns, that is, if $A \\in {\\mathcal M}(n,n)$\n",
    "for some $n$.\n",
    "\n",
    "__Inverses.__\n",
    "Given a square matrix $A\\in {\\mathcal M}(n,n)$, $A^{-1}$ is the _inverse_ \n",
    "of $A$ if $A^{-1}A = {\\mathbf 1}_n$.\n",
    "\n",
    "Not every square matrix has an inverse; the zero matrix ${\\mathbf 0}_n$ is an example.\n",
    "If $A$ has an inverse, $A$ is _invertible_.\n",
    "A square matrix that is not invertible is _singular_.\n",
    "\n",
    "Properties of inverses:\n",
    "+ If $A$ has an inverse, the inverse is unique\n",
    "+ $AA^{-1} = A^{-1}A = {\\mathbf 1}_n$.\n",
    "+ $(A^{-1})^{-1} = A$.\n",
    "+ $(A')^{-1} = (A^{-1})'$.\n",
    "+ If $A$ is invertible, $(\\alpha A)^{-1} = \\alpha^{-1} A^{-1}$ for all nonzero $\\alpha \\in \\Re$.\n",
    "+ If $A$ and $B$ are invertible and have the same dimension, $(AB)^{-1} = B^{-1}A^{-1}$.\n",
    "\n",
    "\n",
    "__Linear Independence and Rank.__\n",
    "Let $\\{a_j \\} \\in \\Re^n$ be a collection of vectors.\n",
    "Recall that $\\{a_j \\}$ are _linearly independent_ if\n",
    "$\\sum_j \\alpha_j a_j = {\\mathbf 0}_n$ implies that $\\alpha_j = 0$ for all $j$.\n",
    "\n",
    "The _row rank_ of a matrix $A \\in {\\mathcal M}(m,n)$ is the maximal number of linearly independent rows it has;\n",
    "equivalently, it is the dimension of the subspace of $\\Re^m$ spanned by the rows of $A$.\n",
    "The _column rank_ of $A$ is to the maximal number of linearly independent columns it has;\n",
    "equivalently, it is the dimension of the subspace of $\\Re^n$ spanned by the columns of $A$.\n",
    "The row rank and the column rank if any matrix are equal; we denote them ${\\mbox{rank}}(A)$.\n",
    "\n",
    "If $A \\in {\\mathcal M}(m,n)$ then ${\\mbox{rank}}(A) \\le \\min(m, n)$.\n",
    "\n",
    "If $A$ and $B$ can be multiplied, ${\\mbox{rank}}(AB) \\le \\min({\\mbox{rank}}(A), {\\mbox{rank}}(B))$.\n",
    "\n",
    "If $A$ and $B$ have the same dimensions, ${\\mbox{rank}}(A+B) \\le {\\mbox{rank}}(A) + {\\mbox{rank}}(B)$.\n",
    "\n",
    "A square matrix $A \\in {\\mathcal M}(n,n)$ is invertible if and only if ${\\mbox{rank}}(A) = n$.\n",
    "\n",
    "\n",
    "#### Definitions\n",
    "\n",
    "Let $A \\in {\\mathcal M}(n,n)$.\n",
    "\n",
    "+ If $A^{-1} = A'$, then $A$ is _orthogonal_.\n",
    "+ If $A_{ij} = 0$ for all $i \\ne j$, $A$ is _diagonal_.\n",
    "+ If $x'Ax \\ge 0$ for all $x \\in \\Re^n$, $A$ is _nonnegative definite_ (or _positive semi-definite_)\n",
    "+ If $x'Ax > 0$ for all $x \\in \\Re^n$ except $x = {\\mathbf 0}$, $A$ is _positive definite_\n",
    "+ $A$ has _full rank_ if ${\\mbox{rank}}(A) = n$. Otherwise, $A$ is _rank deficient_.\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "1. Show that if $A \\in {\\mathcal M}(m,n)$ and $B \\in {\\mathcal M}(n,p)$ then $(AB)' = B'A'$.\n",
    "1. Show that the identity matrix is its own inverse.\n",
    "1. Show that if $A$ is invertible, its inverse is unique (a matrix has at most one inverse).\n",
    "1. Show that if $A$ is invertible, then $A^{-1}A = AA^{-1}$.\n",
    "1. Show that if $A$ and $B$ are both invertible matrices with the same dimensions, then $(AB)^{-1} = B^{-1}A^{-1}$.\n",
    "1. Show that one row of a square matrix $A$ is a multiple of another row of $A$, then $A$ is rank deficient.\n",
    "1. Show that if a square matrix is rank deficient, it is not invertible.\n",
    "1. Show that if $A$ is a diagonal matrix with strictly positive diagonal elements, then $A$ is positive definite.\n",
    "1. Show that the row rank of a matrix is equal to its column rank."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Dot (inner) Products and the Euclidean norm.__\n",
    "\n",
    "If $a, b \\in \\Re^n$ then the _dot product_ of $a$ and $b$ is\n",
    "\\begin{equation*} a \\cdot b = a^Tb = a'b = a_1b_1 + a_2b_2 + \\cdots + a_nb_n.\\end{equation*}\n",
    "\n",
    "If $a \\cdot b = 0$, we say $a$ and $b$ are _orthogonal_, and we write $a \\perp b$.\n",
    "\n",
    "The _Euclidean norm_ or _2-norm_ of $a$ is\n",
    "\\begin{equation*}  \\|a \\|_2 = \\|a \\| = |a| \\equiv \\sqrt{a \\cdot a}.\\end{equation*}\n",
    "(The subscript 2 is often omitted for the 2-norm.)\n",
    "\n",
    "For $1 \\le p < \\infty$, the _$p$-norm_ of $a$ is\n",
    "\\begin{equation*}  \\|a \\|_p \\equiv \\left ( \\sum_{i=1}^n \\left |a_i \\right |^p \\right )^{1/p}.\\end{equation*}\n",
    "\n",
    "The _infinity norm_ of $a$ is\n",
    "\\begin{equation*}  \\|a \\|_\\infty \\equiv \\max_{i=1}^n \\left | a_i \\right |.\\end{equation*}\n",
    "\n",
    "__Outer Products.__\n",
    "\n",
    "If $a, b \\in \\Re^n$ then the _outer product_ of $a$ and $b$ is the matrix\n",
    "$ab \\in {\\mathcal M}(n,n)$ with elements $(ab)_{ij} = a_ib_j$:\n",
    "\\begin{equation*} \n",
    "ab = \n",
    "\\begin{pmatrix} \n",
    "a_1b_1 & a_1b_2 & \\cdots & a_1b_n \\\\\n",
    "a_2b_2 & a_2b_2 & \\cdots & a_2b_n \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "a_nb_1 & a_nb_2 & \\cdots & a_nb_n\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "1. Show that if $R \\in {\\mathcal M}(n,n)$ is an orthogonal matrix, then for\n",
    "all $x \\in \\Re^n$, $\\|Rx\\|_2 = \\|x\\|_2$.\n",
    "1. Show that if $G\\in {\\mathcal M}(n,n) $ is positive definite, then $\\| \\cdot \\|_G$ is a norm on $\\Re^n$, \n",
    "where $\\|x\\|_G \\equiv \\sqrt{x'Gx}$.\n",
    "1. Show that the dot product is an inner product (over the real numbers) on $\\Re^n$.\n",
    "1. Show that the 2-norm $ \\|a \\| \\equiv \\sqrt{a \\cdot a}$ is a norm on $\\Re^n$.\n",
    "1. Show that the 1-norm $\\|a\\|_1 \\equiv \\sum_{i=1}^n | a_{i1} |$ is a norm on $\\Re^n$.\n",
    "1. Show that $\\lim_{p \\rightarrow \\infty} \\|a\\|_p = \\|a\\|_\\infty$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## R interlude\n",
    "Let's look at all these entities and operations in R."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"A\"\n"
     ]
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       "\t<tr><td> 3</td><td> 7</td><td>11</td></tr>\n",
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       "\\begin{tabular}{lll}\n",
       "\t 1 & 5 & 9\\\\\n",
       "\t  2 &  6 & 10\\\\\n",
       "\t  3 &  7 & 11\\\\\n",
       "\t  4 &  8 & 12\\\\\n",
       "\\end{tabular}\n"
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      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "6. 6\n",
       "7. 7\n",
       "8. 8\n",
       "9. 9\n",
       "10. 10\n",
       "11. 11\n",
       "12. 12\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3]\n",
       "[1,]    1    5    9\n",
       "[2,]    2    6   10\n",
       "[3,]    3    7   11\n",
       "[4,]    4    8   12"
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       "\\begin{tabular}{lll}\n",
       "\t 1 & 5 & 9\\\\\n",
       "\t  2 &  6 & 10\\\\\n",
       "\t  3 &  7 & 11\\\\\n",
       "\t  4 &  8 & 12\\\\\n",
       "\\end{tabular}\n"
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      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "6. 6\n",
       "7. 7\n",
       "8. 8\n",
       "9. 9\n",
       "10. 10\n",
       "11. 11\n",
       "12. 12\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3]\n",
       "[1,]    1    5    9\n",
       "[2,]    2    6   10\n",
       "[3,]    3    7   11\n",
       "[4,]    4    8   12"
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     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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       "\t<tr><td> 3</td><td> 7</td><td>11</td></tr>\n",
       "\t<tr><td> 4</td><td> 8</td><td>12</td></tr>\n",
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       "\\begin{tabular}{lll}\n",
       "\t 1 & 5 & 9\\\\\n",
       "\t  2 &  6 & 10\\\\\n",
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       "\t  4 &  8 & 12\\\\\n",
       "\\end{tabular}\n"
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      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "6. 6\n",
       "7. 7\n",
       "8. 8\n",
       "9. 9\n",
       "10. 10\n",
       "11. 11\n",
       "12. 12\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3]\n",
       "[1,]    1    5    9\n",
       "[2,]    2    6   10\n",
       "[3,]    3    7   11\n",
       "[4,]    4    8   12"
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     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# Create a matrix\n",
    "A <- 1:12;  # A is a 1-dimensional array of the integers 1-12\n",
    "dim(A) <- c(4,3);  # Change the dimension of A to be 4 by 3\n",
    "print(\"A\")\n",
    "A  # the matrix A, which is 4 by 3\n",
    "# Another way to do the same thing\n",
    "A <- matrix( \n",
    "+   c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), # data\n",
    "    nrow=4,              # number of rows \n",
    "    ncol=3,              # number of columns \n",
    "    byrow = FALSE) ;     # read the data by column\n",
    "A\n",
    "# Another way\n",
    "A <- matrix( \n",
    "    seq(1,12), # data\n",
    "    nrow=4,              # number of rows \n",
    "    ncol=3,              # number of columns \n",
    "    byrow = FALSE) ;     # read the data by column\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"Transpose of A\"\n"
     ]
    },
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       "1. 1\n",
       "2. 5\n",
       "3. 9\n",
       "4. 2\n",
       "5. 6\n",
       "6. 10\n",
       "7. 3\n",
       "8. 7\n",
       "9. 11\n",
       "10. 4\n",
       "11. 8\n",
       "12. 12\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4]\n",
       "[1,]    1    2    3    4\n",
       "[2,]    5    6    7    8\n",
       "[3,]    9   10   11   12"
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     "execution_count": 3,
     "metadata": {},
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       "1. 1\n",
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       "7. 3\n",
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       "10. 4\n",
       "11. 8\n",
       "12. 12\n",
       "\n",
       "\n"
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       "[3,]    9   10   11   12"
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     "execution_count": 3,
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     "output_type": "execute_result"
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       "1. 1\n",
       "2. 5\n",
       "3. 9\n",
       "4. 2\n",
       "5. 6\n",
       "6. 10\n",
       "7. 3\n",
       "8. 7\n",
       "9. 11\n",
       "10. 4\n",
       "11. 8\n",
       "12. 12\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4]\n",
       "[1,]    1    2    3    4\n",
       "[2,]    5    6    7    8\n",
       "[3,]    9   10   11   12"
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     "execution_count": 3,
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   "source": [
    "# Transpose A\n",
    "print(\"Transpose of A\")\n",
    "t(A) # the transpose of A, which is 3 by 4\n",
    "A <- t(A);  # overwrite A with its transpose: A is now a 3 by 4 array\n",
    "A\n",
    "# Another way to make this matrix\n",
    "A <- matrix( \n",
    "    seq(1,12), # data\n",
    "    nrow=3,              # number of rows \n",
    "    ncol=4,              # number of columns \n",
    "    byrow = TRUE) ;      # read the data by row\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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       "\\begin{tabular}{llll}\n",
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       "1. 1\n",
       "2. 5\n",
       "3. 9\n",
       "4. 2\n",
       "5. 6\n",
       "6. 10\n",
       "7. 3\n",
       "8. 7\n",
       "9. 11\n",
       "10. 4\n",
       "11. 8\n",
       "12. 12\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4]\n",
       "[1,]    1    2    3    4\n",
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       "[3,]    9   10   11   12"
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     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
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    {
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       "\t<tr><td>27</td><td>30</td><td>33</td><td>36</td></tr>\n",
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       "\\end{tabular}\n"
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       "1. 3\n",
       "2. 15\n",
       "3. 27\n",
       "4. 6\n",
       "5. 18\n",
       "6. 30\n",
       "7. 9\n",
       "8. 21\n",
       "9. 33\n",
       "10. 12\n",
       "11. 24\n",
       "12. 36\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4]\n",
       "[1,]    3    6    9   12\n",
       "[2,]   15   18   21   24\n",
       "[3,]   27   30   33   36"
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     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# Scalar multiplication\n",
    "alpha <- 3;\n",
    "A\n",
    "alpha*A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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       "1. -1\n",
       "2. -5\n",
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       "5. -6\n",
       "6. -10\n",
       "7. -3\n",
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       "10. -4\n",
       "11. -8\n",
       "12. -12\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4]\n",
       "[1,]   -1   -2   -3   -4\n",
       "[2,]   -5   -6   -7   -8\n",
       "[3,]   -9  -10  -11  -12"
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     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
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       "1. -1\n",
       "2. -5\n",
       "3. -9\n",
       "4. -2\n",
       "5. -6\n",
       "6. -10\n",
       "7. -3\n",
       "8. -7\n",
       "9. -11\n",
       "10. -4\n",
       "11. -8\n",
       "12. -12\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4]\n",
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       "[2,]   -5   -6   -7   -8\n",
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     "execution_count": 5,
     "metadata": {},
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   "source": [
    "# R understands matrix negation\n",
    "-A\n",
    "-1*A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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       "\\begin{tabular}{llll}\n",
       "\t 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0\\\\\n",
       "\\end{tabular}\n"
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       "1. 0\n",
       "2. 0\n",
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       "5. 0\n",
       "6. 0\n",
       "7. 0\n",
       "8. 0\n",
       "9. 0\n",
       "10. 0\n",
       "11. 0\n",
       "12. 0\n",
       "\n",
       "\n"
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      "text/plain": [
       "     [,1] [,2] [,3] [,4]\n",
       "[1,]    0    0    0    0\n",
       "[2,]    0    0    0    0\n",
       "[3,]    0    0    0    0"
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     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
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    {
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       "\\begin{tabular}{llll}\n",
       "\t 1 & 2 & 3 & 4\\\\\n",
       "\t 5 & 6 & 7 & 8\\\\\n",
       "\t  9 & 10 & 11 & 12\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 5\n",
       "3. 9\n",
       "4. 2\n",
       "5. 6\n",
       "6. 10\n",
       "7. 3\n",
       "8. 7\n",
       "9. 11\n",
       "10. 4\n",
       "11. 8\n",
       "12. 12\n",
       "\n",
       "\n"
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       "[1,]    1    2    3    4\n",
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       "[3,]    9   10   11   12"
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     "execution_count": 6,
     "metadata": {},
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   "source": [
    "# in R, you can create an m by n zero matrix as follows:\n",
    "Z <- matrix(0, 3, 4);\n",
    "Z\n",
    "A + Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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       "\t<tr><td>0</td><td>0</td><td>1</td></tr>\n",
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       "\\begin{tabular}{lll}\n",
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       "\t 0 & 0 & 1\\\\\n",
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       "1. 1\n",
       "2. 0\n",
       "3. 0\n",
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       "7. 0\n",
       "8. 0\n",
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       "\n",
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       "[3,]    0    0    1"
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     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
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       "\t<tr><td>0</td><td>0</td><td>1</td></tr>\n",
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       "\\begin{tabular}{lll}\n",
       "\t 1 & 0 & 0\\\\\n",
       "\t 0 & 1 & 0\\\\\n",
       "\t 0 & 0 & 1\\\\\n",
       "\\end{tabular}\n"
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       "1. 1\n",
       "2. 0\n",
       "3. 0\n",
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       "5. 1\n",
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       "\n",
       "\n"
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       "     [,1] [,2] [,3]\n",
       "[1,]    1    0    0\n",
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     "execution_count": 7,
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   "source": [
    "# To create an n by n identity matrix in R, make a diagonal matrix of ones:\n",
    "I = diag(1,3) # a diagonal matrix of 1s, with dimension 3 by 3\n",
    "I\n",
    "I2 = diag(c(1,1,1)) # a diagonal matrix with the vector (1,1,1) along the diagonal\n",
    "I2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>4</td><td>5</td><td>6</td><td>7</td></tr>\n",
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       "\\begin{tabular}{llll}\n",
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      "text/markdown": [
       "1. 4\n",
       "2. 8\n",
       "3. 12\n",
       "4. 5\n",
       "5. 9\n",
       "6. 13\n",
       "7. 6\n",
       "8. 10\n",
       "9. 14\n",
       "10. 7\n",
       "11. 11\n",
       "12. 15\n",
       "\n",
       "\n"
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       "[1,]    4    5    6    7\n",
       "[2,]    8    9   10   11\n",
       "[3,]   12   13   14   15"
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     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# R supports scalar addition; this is not a formal operation in linear algebra\n",
    "alpha + A;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "6. 10\n",
       "7. 3\n",
       "8. 7\n",
       "9. 11\n",
       "10. 4\n",
       "11. 8\n",
       "12. 12\n",
       "\n",
       "\n"
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       "[1,]    1    2    3    4\n",
       "[2,]    5    6    7    8\n",
       "[3,]    9   10   11   12"
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     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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    {
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       "\n",
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     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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       "1. 25\n",
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       "7. 25\n",
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       "9. 25\n",
       "10. 25\n",
       "11. 25\n",
       "12. 25\n",
       "\n",
       "\n"
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Matrix addition\n",
    "B <- 24:13;  # B is an array with the integers 24 to 13 in descending order \n",
    "dim(B) = c(4,3); # B is now a 4 by 3 array\n",
    "B <- t(B); # B is now a 3 by 4 array\n",
    "A # 1:12\n",
    "B # 24:13\n",
    "A+B # a 3 by 4 array; all entries are 25"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
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       "</ol>\n"
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       "\\begin{enumerate*}\n",
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       "\\item 2\n",
       "\\item 3\n",
       "\\item 4\n",
       "\\end{enumerate*}\n"
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       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "\n",
       "\n"
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       "[1] 1 2 3 4"
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     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
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    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>30</td></tr>\n",
       "\t<tr><td>70</td></tr>\n",
       "\t<tr><td>110</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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       "\\begin{tabular}{l}\n",
       "\t 30\\\\\n",
       "\t 70\\\\\n",
       "\t 110\\\\\n",
       "\\end{tabular}\n"
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       "1. 30\n",
       "2. 70\n",
       "3. 110\n",
       "\n",
       "\n"
      ],
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       "     [,1]\n",
       "[1,]   30\n",
       "[2,]   70\n",
       "[3,]  110"
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     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the (3,1) entry should be 110"
     ]
    }
   ],
   "source": [
    "# Matrix multiplication\n",
    "C <- 1:4;\n",
    "C\n",
    "A%*%C  # a 3 by 2 matrix\n",
    "cat(\"the (3,1) entry should be\", 9*1+10*2+11*3+12*4)  # Check the (3,1) entry"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>220</td><td>180</td><td>140</td></tr>\n",
       "\t<tr><td>580</td><td>476</td><td>372</td></tr>\n",
       "\t<tr><td>940</td><td>772</td><td>604</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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       "\\begin{tabular}{lll}\n",
       "\t 220 & 180 & 140\\\\\n",
       "\t 580 & 476 & 372\\\\\n",
       "\t 940 & 772 & 604\\\\\n",
       "\\end{tabular}\n"
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       "1. 220\n",
       "2. 580\n",
       "3. 940\n",
       "4. 180\n",
       "5. 476\n",
       "6. 772\n",
       "7. 140\n",
       "8. 372\n",
       "9. 604\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3]\n",
       "[1,]  220  180  140\n",
       "[2,]  580  476  372\n",
       "[3,]  940  772  604"
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     "execution_count": 11,
     "metadata": {},
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       "\t 268 & 253 & 238 & 223\\\\\n",
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       "\t 388 & 367 & 346 & 325\\\\\n",
       "\t 448 & 424 & 400 & 376\\\\\n",
       "\\end{tabular}\n"
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       "1. 268\n",
       "2. 328\n",
       "3. 388\n",
       "4. 448\n",
       "5. 253\n",
       "6. 310\n",
       "7. 367\n",
       "8. 424\n",
       "9. 238\n",
       "10. 292\n",
       "11. 346\n",
       "12. 400\n",
       "13. 223\n",
       "14. 274\n",
       "15. 325\n",
       "16. 376\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4]\n",
       "[1,]  268  253  238  223\n",
       "[2,]  328  310  292  274\n",
       "[3,]  388  367  346  325\n",
       "[4,]  448  424  400  376"
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     "execution_count": 11,
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   "source": [
    "# now multiplying a matrix by a matrix  The matrix multiplication operator is %*%\n",
    "A %*% t(B)  # a 3 by 3 matrix\n",
    "t(A) %*% B  # a 4 by 4 matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
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    {
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       "\t<tr><td> 9</td><td>10</td><td>11</td><td>12</td></tr>\n",
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       "1. 1\n",
       "2. 5\n",
       "3. 9\n",
       "4. 2\n",
       "5. 6\n",
       "6. 10\n",
       "7. 3\n",
       "8. 7\n",
       "9. 11\n",
       "10. 4\n",
       "11. 8\n",
       "12. 12\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4]\n",
       "[1,]    1    2    3    4\n",
       "[2,]    5    6    7    8\n",
       "[3,]    9   10   11   12"
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     "execution_count": 12,
     "metadata": {},
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       "\t<tr><td> 9</td><td>10</td><td>11</td><td>12</td></tr>\n",
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       "\\begin{tabular}{llll}\n",
       "\t 1 & 2 & 3 & 4\\\\\n",
       "\t 5 & 6 & 7 & 8\\\\\n",
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       "\\end{tabular}\n"
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       "1. 1\n",
       "2. 5\n",
       "3. 9\n",
       "4. 2\n",
       "5. 6\n",
       "6. 10\n",
       "7. 3\n",
       "8. 7\n",
       "9. 11\n",
       "10. 4\n",
       "11. 8\n",
       "12. 12\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4]\n",
       "[1,]    1    2    3    4\n",
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    "# Multiplication by the identity matrix\n",
    "I3 <- diag(1,3); # I3 is the 3 by 3 identity matrix\n",
    "I3 %*% A  # multiplying from the left by the 3 by 3 identity matrix gives us back A\n",
    "#\n",
    "I4 <- diag(1,4); # I4 is the 4 by 4 identity matrix\n",
    "A %*% I4 # multiplying from the right by the 4 by 4 identity matrix gives us back A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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       "\t<tr><td>1</td><td>8</td><td>9</td><td>8</td></tr>\n",
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       "\t<tr><td>27</td><td>20</td><td>11</td><td>48</td></tr>\n",
       "</tbody>\n",
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       "\\begin{tabular}{llll}\n",
       "\t 1 & 8 & 9 & 8\\\\\n",
       "\t 10 &  6 & 28 & 24\\\\\n",
       "\t 27 & 20 & 11 & 48\\\\\n",
       "\\end{tabular}\n"
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       "1. 1\n",
       "2. 10\n",
       "3. 27\n",
       "4. 8\n",
       "5. 6\n",
       "6. 20\n",
       "7. 9\n",
       "8. 28\n",
       "9. 11\n",
       "10. 8\n",
       "11. 24\n",
       "12. 48\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4]\n",
       "[1,]    1    8    9    8\n",
       "[2,]   10    6   28   24\n",
       "[3,]   27   20   11   48"
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     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "# R also does elementwise multiplication; again, this is not a formal operation in linear algebra\n",
    "A*C  # R copies C repeatedly to fill out the dimension of A, \n",
    "     # then multiplies elements of A by the corresponding element"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
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       "\t<li>2</li>\n",
       "</ol>\n"
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       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\end{enumerate*}\n"
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       "1. 1\n",
       "2. 1\n",
       "3. 1\n",
       "4. 2\n",
       "5. 2\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 1 1 1 2 2"
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     "execution_count": 14,
     "metadata": {},
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       "\\begin{tabular}{l}\n",
       "\t 24\\\\\n",
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       "24"
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       "     [,1]\n",
       "[1,]   24"
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     "execution_count": 14,
     "metadata": {},
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   ],
   "source": [
    "# The dot product\n",
    "a <- 1:5;\n",
    "b <- c(rep(1,3), rep(2,2));  # new function: define a vector by repeating values\n",
    "b\n",
    "t(a) %*% b\n",
    "# another way to do it\n",
    "sum(a*b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
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      "text/plain": [
       "         [,1]\n",
       "[1,] 7.416198"
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   "source": [
    "# the 2-norm\n",
    "norm(as.matrix(a), type=\"2\")  # the \"norm\" function operates on matrices--not lists. \n",
    "                              # The function \"as.matrix()\" turns the list a into a matrix.\n",
    "# alternative 1\n",
    "sqrt(sum(a*a))\n",
    "# alternative 2\n",
    "sqrt(t(a) %*% a)\n",
    "# check:\n",
    "sqrt(1^2+2^2+3^2+4^2+5^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "15"
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      "text/plain": [
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     "execution_count": 16,
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     "output_type": "execute_result"
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      "text/plain": [
       "[1] 15"
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     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
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     "data": {
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       "15"
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       "[1] 15"
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     "execution_count": 16,
     "metadata": {},
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   ],
   "source": [
    "# the 1-norm\n",
    "norm(as.matrix(a), type=\"o\")\n",
    "# alternative \n",
    "sum(abs(a))\n",
    "# check:\n",
    "1+2+3+4+5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interlude: a bit more computation and R\n",
    "\n",
    "The notation $X \\sim P$ means that the random variable $X$ has the probability distribution $P$.\n",
    "Recall that $U[a, b]$ denotes the uniform distribution on the interval $[a, b]$.\n",
    "\n",
    "Recall that the R function <tt>runif()</tt> generates uniformly distributed pseudo-random numbers between 0 and 1.\n",
    "To generate pseudo-random numbers that are uniformly distributed on the interval $[a, b]$, we can re-scale\n",
    "numbers from the interval $[0,1]$ as follows:\n",
    "\n",
    "If $X \\sim U[0,1]$, then $a+(b-a)X \\sim U[a,b]$.\n",
    "\n",
    "Example: to generate pseudo-random numbers on the interval $[-1, 2]$, we can generate pseudo-random numbers\n",
    "on the interval $[0,1]$, multiply them by 3, and subtract 1 from the result.\n",
    "\n",
    "The R function <tt>runif()</tt> takes options <tt>min</tt> and <tt>max</tt> to set the range in this way."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "Plot with title “Histogram of x”"
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Here's how to plot a histogram in R. You will need this for the exercise below.\n",
    "x <- runif(1000, min=-1, max=2)  # generate pseudorandom numbers uniformly distributed between -1 and 2.\n",
    "hist(x)  # plot a histogram of the results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercises\n",
    "\n",
    "+ Write an R script that computes the $p$-norm for general $p$, two different ways, from scratch. Do not use the \"norm\" function.\n",
    "+ Let $A$ be the 2 by 2 rotation matrix:\n",
    "\\begin{equation*} \n",
    "A(\\theta) = \\begin{pmatrix}\n",
    "\\cos \\theta & -\\sin \\theta \\\\\n",
    "\\sin \\theta & \\cos \\theta\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "    - Prove algebraically that $A(\\theta)$ is an orthogonal matrix.\n",
    "    - Write an R script that checks whether $A(\\theta)$ is orthogonal by computing $A(\\theta)A'(\\theta)$ numerically for 1,000 values of $\\theta$ selected at random, uniformly from $[0, 2\\pi]$:\n",
    "        - For each randomly selected value of $\\theta$, calculate $\\sum_{i=1}^2 \\sum_{j=1}^2 \\left ( (A(\\theta)A'(\\theta))_{ij} - ({\\mathbf 1}_2)_{ij} \\right )^2$. (Note: this is the square of the Frobenius matrix norm of the difference between $AA'$ and $\\mathbf 1$. You can code it from scratch or use the R function <tt>norm()</tt> by setting <tt>type=\"F\"</tt> in the call to <tt>norm</tt>.)\n",
    "        - Find the maximum value of those 1,000 sums of squares.\n",
    "        - Plot a histogram of those 1,000 sums of squares."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Trace, Determinant, Inverse\n",
    "\n",
    "### Trace\n",
    "If $A \\in {\\mathcal M}(n,n)$, the _trace_ of $A$ is the sum of the _diagonal elements_ of $A$:\n",
    "\\begin{equation*}  {\\mbox{trace}}(A) = \\sum_{i=1}^n A_{ii}.\\end{equation*}\n",
    "\n",
    "### Determinants\n",
    "\n",
    "The _determinant_ of a square matrix is a real number; how to compute the determinant\n",
    "of a matrix is described below.\n",
    "\n",
    "Properties of determinants: \n",
    "\n",
    "+ $|A| = |A'|$. \n",
    "+ If $A$ and $B$ can be multiplied, $|AB| = |A|\\cdot |B|$.\n",
    "+ $|{\\mathbf 1}_n| = 1$. \n",
    "+ If $A$ is invertible, $|A^{-1}| = |A|^{-1}$.\n",
    "+ $A$ is invertible if and only if its determinant is nonzero.\n",
    "\n",
    "#### Computing the determinant\n",
    "\n",
    "The determinant of a matrix $A \\in {\\mathcal M}(2,2)$ is\n",
    "\\begin{equation*} \n",
    "|A| = A_{11}A_{22} - A_{12}A_{21}.\n",
    "\\end{equation*}\n",
    "\n",
    "The determinant of a matrix $A \\in {\\mathcal M}(3,3)$ is\n",
    "\\begin{equation*} \n",
    "|A| = \n",
    "A_{11}(A_{22}A_{33} - A_{23}A_{32})\n",
    "- A_{12}(A_{21}A_{33}-A_{23}A_{31}) \n",
    "+ A_{13}(A_{21}A_{32}-A_{22}A_{31}).\n",
    "\\end{equation*}\n",
    "\n",
    "Let $S_n$ be the set of all permutations of ${1, \\ldots, n}$.\n",
    "Define the _signature_ of the permutation $\\pi \\in S_n$, $\\mbox{sgn}(\\pi)$ to be $+1$ if\n",
    "$\\pi$ can be derived from $(1, \\ldots, n)$ by interchanging an even number of pairs of elements\n",
    "and $-1$ if $\\pi$ can be derived from $(1, \\ldots, n)$ by interchanging an even number of \n",
    "pairs of elements.\n",
    "\n",
    "E.g., for $n=4$, $\\mbox{sgn}(1,3,2,4) = -1$, since this permutation can be derived from $(1,2,3,4)$ by swapping\n",
    "$3$ and $2$.\n",
    "\n",
    "The determinant of a matrix $A \\in {\\mathcal M}(n,n)$ is then given by\n",
    "\\begin{equation*} \n",
    "\\left |A \\right | \\equiv \\sum_{\\pi \\in S_n} \\mbox{sgn}(\\pi) \\prod_{i=1}^n A_{i,\\pi_i}.\n",
    "\\end{equation*}\n",
    "\n",
    "\n",
    "### Inverses\n",
    "\n",
    "If $A \\in {\\mathcal M}(n,n)$ is invertible, its inverse is\n",
    "\\begin{equation*} \n",
    "A^{-1} = \\frac{\\mbox{adj}(A)}{|A|},\n",
    "\\end{equation*}\n",
    "where $\\mbox{adj}(A)$ is the _adjugate_ of $A$, the transpose of the _matrix of co-factors_.\n",
    "The $(i,j)$ element of the co-factor matrix of $A$ is the determinant of the matrix that\n",
    "results from deleting the $i$th row and $j$th column of $A$, multiplied by $(-1)^{i+j}$.\n",
    "\n",
    "For instance, if $A \\in {\\mathcal M}(2,2)$, its adjugate is\n",
    "\\begin{equation*} \n",
    "\\mbox{adj}(A) \\equiv \n",
    "\\begin{pmatrix}\n",
    "A_{22} & -A_{12} \\\\\n",
    "-A_{21} & A_{11}\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "If $A \\in {\\mathcal M}(3,3)$, its adjugate is\n",
    "\\begin{equation*} \n",
    "\\mbox{adj}(A) \\equiv \n",
    "\\begin{pmatrix}\n",
    " + \\begin{vmatrix} \n",
    "   A_{22} & A_{23} \\\\\n",
    "   A_{32} & A_{33}\n",
    " \\end{vmatrix}\n",
    " &\n",
    " - \\begin{vmatrix}\n",
    "   A_{21} & A_{23} \\\\\n",
    "   A_{31} & A_{33}\n",
    " \\end{vmatrix}\n",
    " &\n",
    " + \\begin{vmatrix}\n",
    "   A_{21} & A_{22} \\\\\n",
    "   A_{31} & A_{32}\n",
    " \\end{vmatrix}\n",
    " \\\\\n",
    " - \\begin{vmatrix} \n",
    "   A_{12} & A_{13} \\\\\n",
    "   A_{32} & A_{33}\n",
    " \\end{vmatrix}\n",
    " &\n",
    " + \\begin{vmatrix}\n",
    "   A_{11} & A_{13} \\\\\n",
    "   A_{31} & A_{33}\n",
    " \\end{vmatrix}\n",
    " &\n",
    " - \\begin{vmatrix}\n",
    "   A_{11} & A_{12} \\\\\n",
    "   A_{31} & A_{32}\n",
    " \\end{vmatrix}\n",
    " \\\\\n",
    " + \\begin{vmatrix} \n",
    "   A_{12} & A_{13} \\\\\n",
    "   A_{22} & A_{23}\n",
    " \\end{vmatrix}\n",
    " &\n",
    " - \\begin{vmatrix}\n",
    "   A_{11} & A_{13} \\\\\n",
    "   A_{21} & A_{23}\n",
    " \\end{vmatrix}\n",
    " &\n",
    " + \\begin{vmatrix}\n",
    "   A_{11} & A_{12} \\\\\n",
    "   A_{21} & A_{22}\n",
    " \\end{vmatrix}\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "#### Examples of computing inverses\n",
    "\n",
    "Let\n",
    "\\begin{equation*} \n",
    "A =\n",
    "\\begin{pmatrix}\n",
    "1 & 2 \\\\\n",
    "6 & 7\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "Then\n",
    "\\begin{equation*} \n",
    "\\mbox{adj}(A) =\n",
    "\\begin{pmatrix}\n",
    "7 & -2 \\\\\n",
    "-6 & 1\n",
    "\\end{pmatrix}\n",
    ",\n",
    "\\end{equation*}\n",
    "and $|A| = 1\\cdot7 - 6\\cdot2 = 7-12 = -5 \\ne 0$, so $A$ is invertible.\n",
    "\\begin{equation*} \n",
    "A^{-1} = \\frac{\\mbox{adj}(A)}{|A|} =\n",
    "-\\frac{1}{5} \n",
    "\\begin{pmatrix}\n",
    "7 & -2 \\\\\n",
    "-6 & 1\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "Let's check the result:\n",
    "\\begin{equation*} \n",
    "AA^{-1} =\n",
    "\\begin{pmatrix}\n",
    "1 & 2 \\\\\n",
    "6 & 7\n",
    "\\end{pmatrix}\n",
    "\\cdot\n",
    "-\\frac{1}{5}\n",
    "\\begin{pmatrix}\n",
    "7 & -2 \\\\\n",
    "-6 & 1\n",
    "\\end{pmatrix}\n",
    "= \n",
    "\\frac{1}{5}\\begin{pmatrix}\n",
    "-7+12 & 2-2 \\\\\n",
    "-42 + 42 & 12-7\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\frac{1}{5}\n",
    "\\begin{pmatrix}\n",
    "5 & 0 \\\\\n",
    "0 & 5\n",
    "\\end{pmatrix}\n",
    "= {\\mathbf 1}_2.\n",
    "\\end{equation*}\n",
    "\n",
    "Let's compute another: take\n",
    "\\begin{equation*} \n",
    "A = \\begin{pmatrix}\n",
    "1 & 2 & 3 \\\\\n",
    "2 & 4 & 6 \\\\\n",
    "3 & 4 & 5\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "Then\n",
    "\\begin{equation*}  \\left |A \\right | = 1 (4 \\cdot 5 - 4 \\cdot 6) -2(2 \\cdot 5 - 3 \\cdot 6) + 3(2 \\cdot 4 - 3 \\cdot 4) = -4 + 16 - 12 = 0.\n",
    "\\end{equation*}\n",
    "Hence, $A$ is not full rank and is not invertible.\n",
    "(The second row is a multiple of the first row, so the rows are linearly dependent.)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
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       "\t<tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr>\n",
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       "\t<tr><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr>\n",
       "\t<tr><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td></tr>\n",
       "\t<tr><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr>\n",
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       "\\begin{tabular}{lllll}\n",
       "\t 1 & 1 & 1 & 1 & 1\\\\\n",
       "\t 2 & 2 & 2 & 2 & 2\\\\\n",
       "\t 3 & 3 & 3 & 3 & 3\\\\\n",
       "\t 4 & 4 & 4 & 4 & 4\\\\\n",
       "\t 5 & 5 & 5 & 5 & 5\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "6. 1\n",
       "7. 2\n",
       "8. 3\n",
       "9. 4\n",
       "10. 5\n",
       "11. 1\n",
       "12. 2\n",
       "13. 3\n",
       "14. 4\n",
       "15. 5\n",
       "16. 1\n",
       "17. 2\n",
       "18. 3\n",
       "19. 4\n",
       "20. 5\n",
       "21. 1\n",
       "22. 2\n",
       "23. 3\n",
       "24. 4\n",
       "25. 5\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4] [,5]\n",
       "[1,]    1    1    1    1    1\n",
       "[2,]    2    2    2    2    2\n",
       "[3,]    3    3    3    3    3\n",
       "[4,]    4    4    4    4    4\n",
       "[5,]    5    5    5    5    5"
      ]
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     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
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     "data": {
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       "15"
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       "[1] 15"
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     "execution_count": 18,
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   "source": [
    "# Trace in R\n",
    "A <- array(1:5,dim=c(5,5))\n",
    "A\n",
    "sum(diag(A))  # diag() extracts the diagonal of a matrix; sum() adds the elements of a list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr>\n",
       "\t<tr><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td></tr>\n",
       "\t<tr><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr>\n",
       "\t<tr><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td></tr>\n",
       "\t<tr><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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       "\\begin{tabular}{lllll}\n",
       "\t 1 & 1 & 1 & 1 & 1\\\\\n",
       "\t 2 & 2 & 2 & 2 & 2\\\\\n",
       "\t 3 & 3 & 3 & 3 & 3\\\\\n",
       "\t 4 & 4 & 4 & 4 & 4\\\\\n",
       "\t 5 & 5 & 5 & 5 & 5\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "6. 1\n",
       "7. 2\n",
       "8. 3\n",
       "9. 4\n",
       "10. 5\n",
       "11. 1\n",
       "12. 2\n",
       "13. 3\n",
       "14. 4\n",
       "15. 5\n",
       "16. 1\n",
       "17. 2\n",
       "18. 3\n",
       "19. 4\n",
       "20. 5\n",
       "21. 1\n",
       "22. 2\n",
       "23. 3\n",
       "24. 4\n",
       "25. 5\n",
       "\n",
       "\n"
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       "     [,1] [,2] [,3] [,4] [,5]\n",
       "[1,]    1    1    1    1    1\n",
       "[2,]    2    2    2    2    2\n",
       "[3,]    3    3    3    3    3\n",
       "[4,]    4    4    4    4    4\n",
       "[5,]    5    5    5    5    5"
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     "execution_count": 19,
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       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.44494921</td><td>0.52317376</td><td>0.03632915</td><td>0.32502890</td><td>0.11407382</td></tr>\n",
       "\t<tr><td>0.1517755</td><td>0.8576347</td><td>0.6506324</td><td>0.7805578</td><td>0.1062984</td></tr>\n",
       "\t<tr><td>0.7462715</td><td>0.4619980</td><td>0.1625752</td><td>0.6339368</td><td>0.6016028</td></tr>\n",
       "\t<tr><td>0.60668472</td><td>0.64195808</td><td>0.09207884</td><td>0.71420528</td><td>0.62334836</td></tr>\n",
       "\t<tr><td>0.79192360</td><td>0.03372087</td><td>0.35195751</td><td>0.07350370</td><td>0.58952493</td></tr>\n",
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       "</table>\n"
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       "\\begin{tabular}{lllll}\n",
       "\t 0.44494921 & 0.52317376 & 0.03632915 & 0.32502890 & 0.11407382\\\\\n",
       "\t 0.1517755 & 0.8576347 & 0.6506324 & 0.7805578 & 0.1062984\\\\\n",
       "\t 0.7462715 & 0.4619980 & 0.1625752 & 0.6339368 & 0.6016028\\\\\n",
       "\t 0.60668472 & 0.64195808 & 0.09207884 & 0.71420528 & 0.62334836\\\\\n",
       "\t 0.79192360 & 0.03372087 & 0.35195751 & 0.07350370 & 0.58952493\\\\\n",
       "\\end{tabular}\n"
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       "1. 0.444949207361788\n",
       "2. 0.151775513775647\n",
       "3. 0.74627153784968\n",
       "4. 0.606684715487063\n",
       "5. 0.791923599550501\n",
       "6. 0.523173755733296\n",
       "7. 0.857634677318856\n",
       "8. 0.461997991194949\n",
       "9. 0.641958083258942\n",
       "10. 0.0337208735290915\n",
       "11. 0.0363291460089386\n",
       "12. 0.650632397271693\n",
       "13. 0.162575213704258\n",
       "14. 0.0920788403600454\n",
       "15. 0.351957513950765\n",
       "16. 0.3250289009884\n",
       "17. 0.780557841993868\n",
       "18. 0.633936773054302\n",
       "19. 0.714205275522545\n",
       "20. 0.073503696359694\n",
       "21. 0.114073819480836\n",
       "22. 0.106298378901556\n",
       "23. 0.6016028188169\n",
       "24. 0.623348356224597\n",
       "25. 0.589524933602661\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "          [,1]       [,2]       [,3]      [,4]      [,5]\n",
       "[1,] 0.4449492 0.52317376 0.03632915 0.3250289 0.1140738\n",
       "[2,] 0.1517755 0.85763468 0.65063240 0.7805578 0.1062984\n",
       "[3,] 0.7462715 0.46199799 0.16257521 0.6339368 0.6016028\n",
       "[4,] 0.6066847 0.64195808 0.09207884 0.7142053 0.6233484\n",
       "[5,] 0.7919236 0.03372087 0.35195751 0.0735037 0.5895249"
      ]
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     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
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       "0.0233555320383095"
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       "0.0233555320383095"
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       "[1] 0.02335553"
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   "source": [
    "# Determinant\n",
    "A <- array(1:5,dim=c(5,5))\n",
    "A  # the rows and columns are multiples of each other, so A is singular (not invertible)\n",
    "det(A) # 0\n",
    "\n",
    "# The set of singular matrices has Lebesgue measure 0, so a random matrix is nonsingular with probability 1\n",
    "A <- array(runif(25), dim=c(5,5)) # 5 by 5 matrix with uniformly distributed elements\n",
    "A\n",
    "det(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 1.6532796</td><td>-0.3517214</td><td> 3.0738478</td><td>-3.0826031</td><td>-0.1338509</td></tr>\n",
       "\t<tr><td> 1.86271860</td><td> 0.08329378</td><td>-5.52354965</td><td> 3.84048152</td><td> 1.20043061</td></tr>\n",
       "\t<tr><td>-0.7561739</td><td> 1.1796194</td><td>-0.2336376</td><td>-0.8476521</td><td> 1.0683307</td></tr>\n",
       "\t<tr><td>-1.5079462</td><td> 0.3122524</td><td> 6.2728416</td><td>-3.5774260</td><td>-2.3831924</td></tr>\n",
       "\t<tr><td>-1.6879742</td><td>-0.2754759</td><td>-4.4558594</td><td> 4.8733703</td><td> 1.4667515</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lllll}\n",
       "\t  1.6532796 & -0.3517214 &  3.0738478 & -3.0826031 & -0.1338509\\\\\n",
       "\t  1.86271860 &  0.08329378 & -5.52354965 &  3.84048152 &  1.20043061\\\\\n",
       "\t -0.7561739 &  1.1796194 & -0.2336376 & -0.8476521 &  1.0683307\\\\\n",
       "\t -1.5079462 &  0.3122524 &  6.2728416 & -3.5774260 & -2.3831924\\\\\n",
       "\t -1.6879742 & -0.2754759 & -4.4558594 &  4.8733703 &  1.4667515\\\\\n",
       "\\end{tabular}\n"
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      "text/markdown": [
       "1. 1.65327956312686\n",
       "2. 1.86271860051819\n",
       "3. -0.75617393583615\n",
       "4. -1.50794620876496\n",
       "5. -1.68797420684956\n",
       "6. -0.351721408805134\n",
       "7. 0.083293776946607\n",
       "8. 1.17961944763411\n",
       "9. 0.312252419380156\n",
       "10. -0.275475863327393\n",
       "11. 3.07384781731378\n",
       "12. -5.52354964637256\n",
       "13. -0.233637629852356\n",
       "14. 6.27284163844156\n",
       "15. -4.45585943343212\n",
       "16. -3.08260305359346\n",
       "17. 3.84048151854076\n",
       "18. -0.84765209895088\n",
       "19. -3.57742598391199\n",
       "20. 4.87337025045763\n",
       "21. -0.133850935756109\n",
       "22. 1.20043061160049\n",
       "23. 1.06833074800049\n",
       "24. -2.38319241035093\n",
       "25. 1.46675147034493\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "           [,1]        [,2]       [,3]       [,4]       [,5]\n",
       "[1,]  1.6532796 -0.35172141  3.0738478 -3.0826031 -0.1338509\n",
       "[2,]  1.8627186  0.08329378 -5.5235496  3.8404815  1.2004306\n",
       "[3,] -0.7561739  1.17961945 -0.2336376 -0.8476521  1.0683307\n",
       "[4,] -1.5079462  0.31225242  6.2728416 -3.5774260 -2.3831924\n",
       "[5,] -1.6879742 -0.27547586 -4.4558594  4.8733703  1.4667515"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Inverse\n",
    "solve(A)  # this is almost never the right thing to do"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eigenvalue / Eigenvector decompositions\n",
    "\n",
    "Let $A \\in {\\mathcal M}(n,n)$ be a square matrix.\n",
    "An _eigenvector_ of $A$ is a non-zero vector $v \\in \\Re^n$ such that, for some scalar $\\lambda \\in \\Re$, $\\lambda \\ne 0$,\n",
    "$Av = \\lambda v$.\n",
    "That is, the effect of multiplying $v$ by the matrix $A$ is to \"stretch or shrink\" $v$ by the scalar\n",
    "$\\lambda$, without changing the direction of $v$ (except perhaps to change its sign).\n",
    "The scalar $\\lambda$ is the _eigenvalue_ associated with the eigenvector $v$.\n",
    "\n",
    "If $A$ is a diagonal matrix, its eigenvectors are the coordinate vectors $e_1 = (1, 0, \\ldots, 0)$, \n",
    "$e_2 = (0, 1, \\ldots, 0)$,\n",
    "$e_n = (0, 0, \\ldots, 1)$, and the associated eigenvalues are the diagonal elements of $A$.\n",
    "That is, the eigenvalue associated with the eigenvector $e_j$ is $A_{jj}$.\n",
    "\n",
    "If $Av = \\lambda v$, then $(A- \\lambda {\\mathbf 1}_n)v = 0$.\n",
    "For that to occur with $v \\ne {\\mathbf 0}_n$, it is necessary that the determinant of $A - \\lambda {\\mathbf 1}_n$\n",
    "be zero.\n",
    "That condition, $\\left |A - \\lambda {\\mathbf 1}_n \\right | = 0$ means that any eigenvalue \n",
    "must be a root of an $n$th degree polynomial in $\\lambda$ \n",
    "called the _characteristic polynomial_ of the matrix $A$.\n",
    "(The leading term in the polynomial is $(-1)^n \\lambda^n$.)\n",
    "It follows that $A$ has at most $n$ real eigenvalues and always has $n$ complex eigenvalues (counting\n",
    "multiple roots as often as they occur).\n",
    "\n",
    "Suppose that $A$ has the $n$ complex eigenvalues $\\{\\lambda_i \\}$\n",
    "(counting multiples as often as they occur).\n",
    "\n",
    "Then:\n",
    "+ $\\mbox{trace}(A) = \\sum_{i=1}^n \\lambda_i$: the trace of $A$ is the sum of its eigenvalues.\n",
    "+ $\\left | A \\right | = \\prod_{i=1}^n \\lambda_i$: the determinant of $A$ is the product of its eigenvalues.\n",
    "+ The eigenvalues of $A^k$ are $\\{\\lambda_i^k\\}_{k=1}^n$.\n",
    "+ If all $\\{\\lambda_i\\}$ are real and non-negative, $A$ is positive semi-definite.\n",
    "+ If all $\\{\\lambda_i\\}$ are real and positive, then $A$ is positive definite.\n",
    "+ If all $\\{\\lambda_i\\}$ are real and nonzero, then $A$ is invertible, and the eigenvalues of $A^{-1}$ are\n",
    "$\\{\\lambda_i^{-1}\\}$.\n",
    "\n",
    "Let $V \\in {\\mathcal M}(n,n)$ be the matrix whose columns are the eigenvectors of a matrix $A$,\n",
    "and let $\\Lambda$ be the diagonal matrix of the corresponding eigenvalues of $A$. Then\n",
    "$A = V \\Lambda V^{-1}$.\n",
    "This factorization is the _eigen-decomposition_ of $A$.\n",
    "\n",
    "Suppose $A$ is positive semi-definite. Then $A$ can be written\n",
    "\\begin{equation*} \n",
    "  A = R \\Lambda R',\n",
    "\\end{equation*}\n",
    "where $R$ is an orthogonal matrix and $\\Lambda$ is the diagonal matrix whose diagonal elements are\n",
    "the eigenvalues of $A$, all of which are non-negative.\n",
    "The columns of $R$ are the eigenvectors of $A$.\n",
    "If $A$ is positive definite, then the diagonal elements of $\\Lambda$ are strictly positive.\n",
    "\n",
    "\n",
    "### Matrix square-roots\n",
    "\n",
    "Suppose that $\\Lambda$ is a diagonal matrix with all non-negative elements.\n",
    "\n",
    "Define the matrix $\\Lambda^{1/2}$ by $(\\Lambda^{1/2})_{ij} = \\sqrt{\\Lambda_{ij}}$:\n",
    "\\begin{equation*} \n",
    "\\Lambda^{1/2} \\equiv \n",
    "\\begin{pmatrix}\n",
    "\\sqrt{\\Lambda_{11}} & 0 & \\cdots & 0 \\\\\n",
    "0 & \\sqrt{\\Lambda_{22}} & \\cdots & 0 \\\\\n",
    "\\vdots & \\vdots & \\ddots & 0 \\\\\n",
    "0 & 0 & \\cdots & \\sqrt{\\Lambda_{nn}}\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "\n",
    "Then $\\Lambda^{1/2}$ is the _matrix square root_ of \n",
    "$\\Lambda$, in the sense that $\\Lambda^{1/2}\\Lambda^{1/2} = \\Lambda$.\n",
    "\n",
    "Suppose $A \\in {\\mathcal M}(n,n)$ is positive semi-definite, with the eigen-decomposition $R\\Lambda R'$,\n",
    "$R$ orthogonal, $\\Lambda$ diagonal with non-negative elements.\n",
    "Define the matrix \n",
    "\\begin{equation*} \n",
    "A^{1/2} \\equiv R \\Lambda^{1/2} R'.\n",
    "\\end{equation*}\n",
    "Then $A^{1/2}$ is the matrix square root of $A$, in the sense that $A^{1/2}A^{1/2} = A$.\n",
    "\n",
    "### Inverses of Positive-Definite Matrices\n",
    "\n",
    "Suppose that $\\Lambda \\in {\\mathcal M}(n,n)$ is a diagonal matrix with strictly positive\n",
    "diagonal elements (i.e., $\\Lambda$ is a diagonal, positive definite matrix).\n",
    "Then the inverse of $\\Lambda$ has elements\n",
    "\\begin{equation*} \n",
    "(\\Lambda^{-1})_{ij} = \n",
    "\\left \\{ \n",
    "\\begin{array}{ll} \n",
    "0, & i \\ne j \\\\ \n",
    "\\Lambda_{ij}^{-1}, & i=j. \n",
    "\\end{array} \n",
    "\\right . \n",
    "\\end{equation*}\n",
    "\n",
    "We can verify this by direct multiplication:\n",
    "\\begin{equation*} \n",
    "(\\Lambda \\Lambda^{-1})_{ij} = \n",
    "\\sum_{\\ell=1}^n \\Lambda_{i\\ell} (\\Lambda^{-1})_{\\ell j}\n",
    "= \\left \\{ \\begin{array}{ll} 0, & i \\ne j \\\\ 1, & i=j. \\end{array} \\right .\n",
    "\\end{equation*}\n",
    "\n",
    "Suppose that $A \\in {\\mathcal M}(n,n)$ is positive definite, with the eigen-decomposition $R\\Lambda R'$.\n",
    "Then $A^{-1} = R \\Lambda^{-1} R'$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "+ Show that if $\\Lambda \\in {\\mathcal M}(n,n)$ is a diagonal matrix with non-negative elements, then \n",
    "the matrix with elements $(\\Lambda^{1/2})_{ij} = \\sqrt{\\Lambda_{ij}}$ is its matrix square root.\n",
    "+ Show that if $A \\in {\\mathcal M}(n,n)$ is positive definite with the eigen-decomposition $R\\Lambda R'$, then\n",
    "$A^{1/2} \\equiv R\\Lambda^{1/2}R'$ is the matrix square root of $A$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$values</dt>\n",
       "\t\t<dd><ol class=list-inline>\n",
       "\t<li>3.65109340893717</li>\n",
       "\t<li>0.726109445035784</li>\n",
       "\t<li>-0.377202853972958</li>\n",
       "</ol>\n",
       "</dd>\n",
       "\t<dt>$vectors</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td>-0.2268408</td><td>-0.8167856</td><td> 0.3520763</td></tr>\n",
       "\t<tr><td>-0.6013762</td><td> 0.2237099</td><td>-0.4848805</td></tr>\n",
       "\t<tr><td>-0.7660874</td><td> 0.5318037</td><td> 0.8005830</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$values] \\begin{enumerate*}\n",
       "\\item 3.65109340893717\n",
       "\\item 0.726109445035784\n",
       "\\item -0.377202853972958\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item[\\$vectors] \\begin{tabular}{lll}\n",
       "\t -0.2268408 & -0.8167856 &  0.3520763\\\\\n",
       "\t -0.6013762 &  0.2237099 & -0.4848805\\\\\n",
       "\t -0.7660874 &  0.5318037 &  0.8005830\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$values\n",
       ":   1. 3.65109340893717\n",
       "2. 0.726109445035784\n",
       "3. -0.377202853972958\n",
       "\n",
       "\n",
       "\n",
       "$vectors\n",
       ":   1. -0.2268408088325\n",
       "2. -0.601376173173819\n",
       "3. -0.766087426986653\n",
       "4. -0.816785593579776\n",
       "5. 0.223709859512343\n",
       "6. 0.531803716494631\n",
       "7. 0.352076317093485\n",
       "8. -0.484880508717436\n",
       "9. 0.800583012065463\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$values\n",
       "[1]  3.6510934  0.7261094 -0.3772029\n",
       "\n",
       "$vectors\n",
       "           [,1]       [,2]       [,3]\n",
       "[1,] -0.2268408 -0.8167856  0.3520763\n",
       "[2,] -0.6013762  0.2237099 -0.4848805\n",
       "[3,] -0.7660874  0.5318037  0.8005830\n"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>3.65109340893717</li>\n",
       "\t<li>0.726109445035784</li>\n",
       "\t<li>-0.377202853972958</li>\n",
       "</ol>\n"
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       "\\begin{enumerate*}\n",
       "\\item 3.65109340893717\n",
       "\\item 0.726109445035784\n",
       "\\item -0.377202853972958\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 3.65109340893717\n",
       "2. 0.726109445035784\n",
       "3. -0.377202853972958\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  3.6510934  0.7261094 -0.3772029"
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     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
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     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>-0.2268408</td><td>-0.8167856</td><td> 0.3520763</td></tr>\n",
       "\t<tr><td>-0.6013762</td><td> 0.2237099</td><td>-0.4848805</td></tr>\n",
       "\t<tr><td>-0.7660874</td><td> 0.5318037</td><td> 0.8005830</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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       "\\begin{tabular}{lll}\n",
       "\t -0.2268408 & -0.8167856 &  0.3520763\\\\\n",
       "\t -0.6013762 &  0.2237099 & -0.4848805\\\\\n",
       "\t -0.7660874 &  0.5318037 &  0.8005830\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. -0.2268408088325\n",
       "2. -0.601376173173819\n",
       "3. -0.766087426986653\n",
       "4. -0.816785593579776\n",
       "5. 0.223709859512343\n",
       "6. 0.531803716494631\n",
       "7. 0.352076317093485\n",
       "8. -0.484880508717436\n",
       "9. 0.800583012065463\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "           [,1]       [,2]       [,3]\n",
       "[1,] -0.2268408 -0.8167856  0.3520763\n",
       "[2,] -0.6013762  0.2237099 -0.4848805\n",
       "[3,] -0.7660874  0.5318037  0.8005830"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Eigen decompositions in R\n",
    "A <- array(c(1,1,1,1,2,3,0,1,1), dim=c(3,3));\n",
    "eigen(A)\n",
    "\n",
    "# eigen() is an example of a function that returns more than one variable. Here's how you access them:\n",
    "decomp <- eigen(A);\n",
    "decomp$values\n",
    "decomp$vectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>-0.2268408088325</li>\n",
       "\t<li>-0.601376173173819</li>\n",
       "\t<li>-0.766087426986653</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item -0.2268408088325\n",
       "\\item -0.601376173173819\n",
       "\\item -0.766087426986653\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. -0.2268408088325\n",
       "2. -0.601376173173819\n",
       "3. -0.766087426986653\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] -0.2268408 -0.6013762 -0.7660874"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>-0.2268408</td></tr>\n",
       "\t<tr><td>-0.6013762</td></tr>\n",
       "\t<tr><td>-0.7660874</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{l}\n",
       "\t -0.2268408\\\\\n",
       "\t -0.6013762\\\\\n",
       "\t -0.7660874\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. -0.2268408088325\n",
       "2. -0.601376173173819\n",
       "3. -0.766087426986653\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "           [,1]\n",
       "[1,] -0.2268408\n",
       "[2,] -0.6013762\n",
       "[3,] -0.7660874"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# let's check whether these really are eigenvectors!\n",
    "ev1 <- decomp$vectors[,1];  # the first column of the eigenvector matrix: an eigenvector\n",
    "ev1\n",
    "A %*% ev1 / decomp$values[1]  # this should give us back the first eigenvector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>1</td><td>1</td><td>0</td></tr>\n",
       "\t<tr><td>1</td><td>2</td><td>1</td></tr>\n",
       "\t<tr><td>1</td><td>3</td><td>1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t 1 & 1 & 0\\\\\n",
       "\t 1 & 2 & 1\\\\\n",
       "\t 1 & 3 & 1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 1\n",
       "3. 1\n",
       "4. 1\n",
       "5. 2\n",
       "6. 3\n",
       "7. 0\n",
       "8. 1\n",
       "9. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3]\n",
       "[1,]    1    1    0\n",
       "[2,]    1    2    1\n",
       "[3,]    1    3    1"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 1.000000e+00</td><td> 1.000000e+00</td><td>-5.551115e-16</td></tr>\n",
       "\t<tr><td>1</td><td>2</td><td>1</td></tr>\n",
       "\t<tr><td>1</td><td>3</td><td>1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t  1.000000e+00 &  1.000000e+00 & -5.551115e-16\\\\\n",
       "\t 1 & 2 & 1\\\\\n",
       "\t 1 & 3 & 1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 1\n",
       "3. 0.999999999999999\n",
       "4. 0.999999999999999\n",
       "5. 2\n",
       "6. 3\n",
       "7. -5.55111512312578e-16\n",
       "8. 0.999999999999999\n",
       "9. 0.999999999999999\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2]          [,3]\n",
       "[1,]    1    1 -5.551115e-16\n",
       "[2,]    1    2  1.000000e+00\n",
       "[3,]    1    3  1.000000e+00"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check that the eigen-decomposition of A actually holds\n",
    "A\n",
    "decomp$vectors %*% diag(decomp$values) %*% solve(decomp$vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "4"
      ],
      "text/latex": [
       "4"
      ],
      "text/markdown": [
       "4"
      ],
      "text/plain": [
       "[1] 4"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "4"
      ],
      "text/latex": [
       "4"
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      "text/markdown": [
       "4"
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      "text/plain": [
       "[1] 4"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check that the sum of the eigenvalues is the trace\n",
    "sum(diag(A))\n",
    "sum(decomp$values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "-1"
      ],
      "text/latex": [
       "-1"
      ],
      "text/markdown": [
       "-1"
      ],
      "text/plain": [
       "[1] -1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "-1"
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       "-1"
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       "-1"
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      "text/plain": [
       "[1] -1"
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     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check that the product of the eigenvalues is the determinant\n",
    "det(A)\n",
    "prod(decomp$values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 1</td><td> 1</td><td>-1</td></tr>\n",
       "\t<tr><td> 1.026956e-15</td><td>-1.000000e+00</td><td> 1.000000e+00</td></tr>\n",
       "\t<tr><td>-1</td><td> 2</td><td>-1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t  1 &  1 & -1\\\\\n",
       "\t  1.026956e-15 & -1.000000e+00 &  1.000000e+00\\\\\n",
       "\t -1 &  2 & -1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 0.999999999999998\n",
       "2. 1.02695629777827e-15\n",
       "3. -1\n",
       "4. 0.999999999999999\n",
       "5. -1\n",
       "6. 2\n",
       "7. -0.999999999999999\n",
       "8. 1\n",
       "9. -1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "              [,1] [,2] [,3]\n",
       "[1,]  1.000000e+00    1   -1\n",
       "[2,]  1.026956e-15   -1    1\n",
       "[3,] -1.000000e+00    2   -1"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 1.000000e+00</td><td>-6.661338e-16</td><td> 1.221245e-15</td></tr>\n",
       "\t<tr><td>-1.110223e-16</td><td> 1.000000e+00</td><td>-4.440892e-16</td></tr>\n",
       "\t<tr><td>8.881784e-16</td><td>1.110223e-15</td><td>1.000000e+00</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t  1.000000e+00 & -6.661338e-16 &  1.221245e-15\\\\\n",
       "\t -1.110223e-16 &  1.000000e+00 & -4.440892e-16\\\\\n",
       "\t 8.881784e-16 & 1.110223e-15 & 1.000000e+00\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 0.999999999999999\n",
       "2. -1.11022302462516e-16\n",
       "3. 8.88178419700125e-16\n",
       "4. -6.66133814775094e-16\n",
       "5. 1\n",
       "6. 1.11022302462516e-15\n",
       "7. 1.22124532708767e-15\n",
       "8. -4.44089209850063e-16\n",
       "9. 0.999999999999999\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "              [,1]          [,2]          [,3]\n",
       "[1,]  1.000000e+00 -6.661338e-16  1.221245e-15\n",
       "[2,] -1.110223e-16  1.000000e+00 -4.440892e-16\n",
       "[3,]  8.881784e-16  1.110223e-15  1.000000e+00"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check whether we can invert A using its eigen-decomposition\n",
    "Ainv <- decomp$vectors %*% diag(1/decomp$values) %*% solve(decomp$vectors);\n",
    "Ainv\n",
    "A %*% Ainv  # this should be the identity matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>-0.2268408</td><td>-0.6013762</td><td>-0.7660874</td></tr>\n",
       "\t<tr><td>-0.8167856</td><td> 0.2237099</td><td> 0.5318037</td></tr>\n",
       "\t<tr><td> 0.3520763</td><td>-0.4848805</td><td> 0.8005830</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t -0.2268408 & -0.6013762 & -0.7660874\\\\\n",
       "\t -0.8167856 &  0.2237099 &  0.5318037\\\\\n",
       "\t  0.3520763 & -0.4848805 &  0.8005830\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. -0.2268408088325\n",
       "2. -0.816785593579776\n",
       "3. 0.352076317093485\n",
       "4. -0.601376173173819\n",
       "5. 0.223709859512343\n",
       "6. -0.484880508717436\n",
       "7. -0.766087426986653\n",
       "8. 0.531803716494631\n",
       "9. 0.800583012065463\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "           [,1]       [,2]       [,3]\n",
       "[1,] -0.2268408 -0.6013762 -0.7660874\n",
       "[2,] -0.8167856  0.2237099  0.5318037\n",
       "[3,]  0.3520763 -0.4848805  0.8005830"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>-0.5152664</td><td>-0.9918796</td><td>-0.3741398</td></tr>\n",
       "\t<tr><td>-1.0057615</td><td>-0.1039075</td><td> 0.3793761</td></tr>\n",
       "\t<tr><td> 0.1750332</td><td>-0.8801187</td><td> 0.6390625</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t -0.5152664 & -0.9918796 & -0.3741398\\\\\n",
       "\t -1.0057615 & -0.1039075 &  0.3793761\\\\\n",
       "\t  0.1750332 & -0.8801187 &  0.6390625\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. -0.515266412477597\n",
       "2. -1.00576151155975\n",
       "3. 0.175033178939936\n",
       "4. -0.991879581156375\n",
       "5. -0.103907534013743\n",
       "6. -0.880118679548961\n",
       "7. -0.374139808809686\n",
       "8. 0.379376112576494\n",
       "9. 0.639062485972922\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "           [,1]       [,2]       [,3]\n",
       "[1,] -0.5152664 -0.9918796 -0.3741398\n",
       "[2,] -1.0057615 -0.1039075  0.3793761\n",
       "[3,]  0.1750332 -0.8801187  0.6390625"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# this matrix is not positive-definite, so the eigenvector matrix is not orthogonal: its\n",
    "# transpose is not its inverse. Let's verify that:\n",
    "t(decomp$vectors)\n",
    "solve(decomp$vectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>3</td><td>6</td><td>2</td></tr>\n",
       "\t<tr><td> 6</td><td>14</td><td> 5</td></tr>\n",
       "\t<tr><td>2</td><td>5</td><td>2</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t 3 & 6 & 2\\\\\n",
       "\t  6 & 14 &  5\\\\\n",
       "\t 2 & 5 & 2\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 3\n",
       "2. 6\n",
       "3. 2\n",
       "4. 6\n",
       "5. 14\n",
       "6. 5\n",
       "7. 2\n",
       "8. 5\n",
       "9. 2\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3]\n",
       "[1,]    3    6    2\n",
       "[2,]    6   14    5\n",
       "[3,]    2    5    2"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>18.4052979838975</li>\n",
       "\t<li>0.481973433502735</li>\n",
       "\t<li>0.11272858259973</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 18.4052979838975\n",
       "\\item 0.481973433502735\n",
       "\\item 0.11272858259973\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 18.4052979838975\n",
       "2. 0.481973433502735\n",
       "3. 0.11272858259973\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 18.4052980  0.4819734  0.1127286"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 1.000000e+00</td><td>-1.110223e-16</td><td>-1.665335e-16</td></tr>\n",
       "\t<tr><td>-1.110223e-16</td><td> 1.000000e+00</td><td>-5.551115e-17</td></tr>\n",
       "\t<tr><td>-1.387779e-16</td><td>-5.551115e-17</td><td> 1.000000e+00</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t  1.000000e+00 & -1.110223e-16 & -1.665335e-16\\\\\n",
       "\t -1.110223e-16 &  1.000000e+00 & -5.551115e-17\\\\\n",
       "\t -1.387779e-16 & -5.551115e-17 &  1.000000e+00\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. -1.11022302462516e-16\n",
       "3. -1.38777878078145e-16\n",
       "4. -1.11022302462516e-16\n",
       "5. 1\n",
       "6. -5.55111512312578e-17\n",
       "7. -1.66533453693773e-16\n",
       "8. -5.55111512312578e-17\n",
       "9. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "              [,1]          [,2]          [,3]\n",
       "[1,]  1.000000e+00 -1.110223e-16 -1.665335e-16\n",
       "[2,] -1.110223e-16  1.000000e+00 -5.551115e-17\n",
       "[3,] -1.387779e-16 -5.551115e-17  1.000000e+00"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now let's try it with a positive-definite matrix. We can make a positive definite matrix from a nonsingular matrix\n",
    "Apd <- t(A) %*% A;\n",
    "Apd\n",
    "decomp <- eigen(Apd);\n",
    "decomp$values  # the eigenvalues should be non-negative\n",
    "decomp$vectors %*% t(decomp$vectors) # and the matrix of eigenvectors should be orthogonal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.3797058</td><td>0.8709960</td><td>0.3117524</td></tr>\n",
       "\t<tr><td> 0.8226923</td><td>-0.1638011</td><td>-0.5443773</td></tr>\n",
       "\t<tr><td> 0.4230850</td><td>-0.4631795</td><td> 0.7787579</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t 0.3797058 & 0.8709960 & 0.3117524\\\\\n",
       "\t  0.8226923 & -0.1638011 & -0.5443773\\\\\n",
       "\t  0.4230850 & -0.4631795 &  0.7787579\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 0.379705772677359\n",
       "2. 0.822692294585154\n",
       "3. 0.423084996928164\n",
       "4. 0.870995956147712\n",
       "5. -0.163801092210104\n",
       "6. -0.463179497133789\n",
       "7. 0.311752418707253\n",
       "8. -0.544377250278694\n",
       "9. 0.778757882020584\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "          [,1]       [,2]       [,3]\n",
       "[1,] 0.3797058  0.8709960  0.3117524\n",
       "[2,] 0.8226923 -0.1638011 -0.5443773\n",
       "[3,] 0.4230850 -0.4631795  0.7787579"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.3797058</td><td>0.8709960</td><td>0.3117524</td></tr>\n",
       "\t<tr><td> 0.8226923</td><td>-0.1638011</td><td>-0.5443773</td></tr>\n",
       "\t<tr><td> 0.4230850</td><td>-0.4631795</td><td> 0.7787579</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t 0.3797058 & 0.8709960 & 0.3117524\\\\\n",
       "\t  0.8226923 & -0.1638011 & -0.5443773\\\\\n",
       "\t  0.4230850 & -0.4631795 &  0.7787579\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 0.379705772677359\n",
       "2. 0.822692294585154\n",
       "3. 0.423084996928164\n",
       "4. 0.870995956147712\n",
       "5. -0.163801092210104\n",
       "6. -0.463179497133789\n",
       "7. 0.311752418707253\n",
       "8. -0.544377250278694\n",
       "9. 0.778757882020584\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "          [,1]       [,2]       [,3]\n",
       "[1,] 0.3797058  0.8709960  0.3117524\n",
       "[2,] 0.8226923 -0.1638011 -0.5443773\n",
       "[3,] 0.4230850 -0.4631795  0.7787579"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the eigenvector matrix should be orthogonal. Let's check:\n",
    "t(decomp$vectors)\n",
    "solve(decomp$vectors) # should be equal to the transpose"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>1.148516</td><td>1.259494</td><td>0.307545</td></tr>\n",
       "\t<tr><td>1.259494</td><td>3.345303</td><td>1.105722</td></tr>\n",
       "\t<tr><td>0.3075450</td><td>1.1057220</td><td>0.8263141</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t 1.148516 & 1.259494 & 0.307545\\\\\n",
       "\t 1.259494 & 3.345303 & 1.105722\\\\\n",
       "\t 0.3075450 & 1.1057220 & 0.8263141\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 1.14851630310121\n",
       "2. 1.25949448307611\n",
       "3. 0.307545035094985\n",
       "4. 1.25949448307611\n",
       "5. 3.34530306281036\n",
       "6. 1.10572196552862\n",
       "7. 0.307545035094985\n",
       "8. 1.10572196552862\n",
       "9. 0.826314096658137\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "         [,1]     [,2]      [,3]\n",
       "[1,] 1.148516 1.259494 0.3075450\n",
       "[2,] 1.259494 3.345303 1.1057220\n",
       "[3,] 0.307545 1.105722 0.8263141"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>3</td><td>6</td><td>2</td></tr>\n",
       "\t<tr><td> 6</td><td>14</td><td> 5</td></tr>\n",
       "\t<tr><td>2</td><td>5</td><td>2</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t 3 & 6 & 2\\\\\n",
       "\t  6 & 14 &  5\\\\\n",
       "\t 2 & 5 & 2\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 3\n",
       "2. 6\n",
       "3. 2\n",
       "4. 6\n",
       "5. 14\n",
       "6. 5\n",
       "7. 2\n",
       "8. 5\n",
       "9. 2\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3]\n",
       "[1,]    3    6    2\n",
       "[2,]    6   14    5\n",
       "[3,]    2    5    2"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# lets make the matrix square root of Apd\n",
    "ApdHalf <- decomp$vectors %*% diag(sqrt(decomp$values)) %*% t(decomp$vectors);\n",
    "ApdHalf\n",
    "ApdHalf %*% ApdHalf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving Linear Systems\n",
    "\n",
    "We will often need to solve linear equations of the form $Ax = b$, where $A \\in {\\mathcal M}(m,n)$ and\n",
    "$b \\in {\\mathcal M}(1,m)$ are known, and $x \\in {\\mathcal M}(n,1)$ is the unknown vector we seek.\n",
    "\n",
    "If the rows of $A$ are linearly independent and $n \\ge m$, there is at least one solution.\n",
    "If $n < m$, there may be no solution.\n",
    "\n",
    "For the moment, suppose $A$ is an invertible square matrix with dimension $n$.\n",
    "Then, formally, we can find $x$ by multiplying both sides by the inverse of $A$:\n",
    "\n",
    "\\begin{equation*} \n",
    "  A^{-1} A x = A^{-1} b.\n",
    "\\end{equation*}\n",
    "Since $A^{-1}A = {\\mathbf 1}_n$, this becomes\n",
    "\\begin{equation*} \n",
    "  {\\mathbf 1}_n x = x = A^{-1} b.\n",
    "\\end{equation*}\n",
    "\n",
    "However, this is not a good way to find $x$: it is unnecessarily numerically unstable (ill conditioned) compared\n",
    "with other ways of solving the linear system.\n",
    "\n",
    "Better methods include matrix factorizations or decompositions (LU, QR, Cholesky), Gaussian elimination, etc.\n",
    "We won't study those methods in detail, but will present Gaussian elimination briefly.\n",
    "\n",
    "__Gaussian elimination.__ \n",
    "Gaussian elimination is a method that can be used to find the rank of a matrix and to solve a linear system.\n",
    "Gaussian elimination involves three operations:\n",
    "\n",
    "+ swap two rows\n",
    "+ multiply a row by a nonzero scalar \n",
    "+ add a scalar times one row to another row\n",
    "\n",
    "All of those operations preserves the linear system:\n",
    "a vector $x$ solves the original system if and only if it solves the transformed system.\n",
    "\n",
    "The goal of Gaussian elimination is to use those operations to transform \n",
    "$A$ into _upper triangular form_, in which all elements \"below\" the diagonal are zero.\n",
    "More specifically, it transforms to _row eschelon form_, in which the first non-zero element of each row is\n",
    "to the right of the first non-zero element of the row above it.\n",
    "It is trivial to solve a linear system (using _back substitution_)\n",
    "once it is in upper-triangular form, as we shall see.\n",
    "\n",
    "Here's an illustration.  Let\n",
    "\\begin{equation*} \n",
    "A = \\begin{pmatrix} \n",
    "2 & 4 & 4 \\\\\n",
    "6 & 1 & 5 \\\\\n",
    "3 & 4 & 3\n",
    "\\end{pmatrix}\n",
    "\\end{equation*}\n",
    "and\n",
    "\\begin{equation*} \n",
    "b = \n",
    "\\begin{pmatrix} \n",
    "2 \\\\\n",
    "1 \\\\\n",
    "1\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "\n",
    "We start by forming the \"augmented\" matrix that consists of appending $b$ as an additional column of $A$:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "2 & 4 & 4 & 2\\\\\n",
    "6 & 1 & 5 & 1\\\\\n",
    "3 & 4 & 3 & 1\n",
    "\\end{pmatrix}\n",
    "\\end{equation*}\n",
    "Divide the first row by 2\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2 & 2 & 1\\\\\n",
    "6 & 1 & 5 & 1\\\\\n",
    "3 & 4 & 3 & 1\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Subtract 6 times the first row from the second row:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2   & 2  &  1\\\\\n",
    "0 & -11 & -7 & -5\\\\\n",
    "3 & 4 & 3 & 1\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Subtract three times the first row from the third row:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2   & 2  &  1\\\\\n",
    "0 & -11 & -7 & -5\\\\\n",
    "0 & -2 & -3 & -2\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Multiply the second and third rows by -1 and swap them:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2   & 2  &  1\\\\\n",
    "0 & 2 & 3 &  2  \\\\\n",
    "0 & 11 & 7 & 5\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Divide the second row by 2:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2 & 2   &  1\\\\\n",
    "0 & 1 & 1.5 &  1  \\\\\n",
    "0 & 11 & 7 & 5\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Subtract 11 times the second row from the third row:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2 & 2   &  1\\\\\n",
    "0 & 1 & 1.5 &  1  \\\\\n",
    "0 & 0 & -9.5 & -6\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Multiply the last row by -1/9.5:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2 & 2   &  1\\\\\n",
    "0 & 1 & 1.5 &  1  \\\\\n",
    "0 & 0 & 1  & 0.63158\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "We now use _back substitution_ to find $x$: \n",
    "\n",
    "+ $x_3 = 0.63158$.\n",
    "+ $x_2 + 1.5\\times 0.63158 = 1$, so $x_2 = 0.05263$.\n",
    "+ $x_1 + 2\\times 0.05263 + 2\\times0.63158 = 1$, so $x_1 = -0.36842$.\n",
    "\n",
    "This gives \n",
    "\\begin{equation*} \n",
    "x = \\begin{pmatrix}\n",
    "-0.36842 \\\\\n",
    " 0.05263 \\\\\n",
    " 0.63158\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "Multiplying by $A$ gives \n",
    "\\begin{equation*} \n",
    "Ax = \\begin{pmatrix} \n",
    "2 & 3 & 4 \\\\\n",
    "6 & 1 & 5 \\\\\n",
    "3 & 4 & 3\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "-0.36842 \\\\\n",
    " 0.05263 \\\\\n",
    " 0.63158\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "2 \\\\\n",
    "1.00001 \\\\\n",
    "1\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>2</td><td>4</td><td>4</td></tr>\n",
       "\t<tr><td>6</td><td>1</td><td>5</td></tr>\n",
       "\t<tr><td>3</td><td>4</td><td>3</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       "\t 2 & 4 & 4\\\\\n",
       "\t 6 & 1 & 5\\\\\n",
       "\t 3 & 4 & 3\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 2\n",
       "2. 6\n",
       "3. 3\n",
       "4. 4\n",
       "5. 1\n",
       "6. 4\n",
       "7. 4\n",
       "8. 5\n",
       "9. 3\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3]\n",
       "[1,]    2    4    4\n",
       "[2,]    6    1    5\n",
       "[3,]    3    4    3"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 2\n",
       "2. 1\n",
       "3. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 2 1 1"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>-0.368421052631579</li>\n",
       "\t<li>0.0526315789473685</li>\n",
       "\t<li>0.631578947368421</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item -0.368421052631579\n",
       "\\item 0.0526315789473685\n",
       "\\item 0.631578947368421\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. -0.368421052631579\n",
       "2. 0.0526315789473685\n",
       "3. 0.631578947368421\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] -0.36842105  0.05263158  0.63157895"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Let's try this in R\n",
    "A <- array(c(2,6,3,4,1,4,4,5,3), dim=c(3,3));\n",
    "A\n",
    "b <- c(2,1,1);\n",
    "b\n",
    "solve(A,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gaussian elimination, continued.\n",
    "What happens if $A$ is rank deficient?\n",
    "\n",
    "Consider the matrix \n",
    "\\begin{equation*} \n",
    "A = \\begin{pmatrix} \n",
    "2 & 4 & 4 \\\\\n",
    "6 & 1 & 5 \\\\\n",
    "10 & 9 & 13\n",
    "\\end{pmatrix}\n",
    ".\n",
    "\\end{equation*}\n",
    "Let's perform Gaussian elimination on it.\n",
    "Divide the first row by 2\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2 & 2 \\\\\n",
    "6 & 1 & 5 \\\\\n",
    "10 & 9 & 13 \n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Subtract 6 times the first row from the second row:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2   & 2  \\\\\n",
    "0 & -11 & -7 \\\\\n",
    "10 & 9 & 13 \n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Subtract ten times the first row from the third row:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2   & 2  \\\\\n",
    "0 & -11 & -7\\\\ \n",
    "0 & -11 & -7 \n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "Subtract the 2nd row from the third row:\n",
    "\\begin{equation*} \n",
    "\\begin{pmatrix} \n",
    "1 & 2   & 2  \\\\\n",
    "0 & -11 & -7\\\\ \n",
    "0 & 0 & 0\n",
    "\\end{pmatrix}.\n",
    "\\end{equation*}\n",
    "The third row is now zero: there are only two nonzero rows once we transform to upper-triangular form.\n",
    "The rank of the matrix $A$ is therefore 2, not 3 (it is rank deficient).\n",
    "This matrix $A$ is not invertible."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>3.28678678551627</li>\n",
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       "\t<li>-3.02500161149644</li>\n",
       "\t<li>1.36130631294013</li>\n",
       "\t<li>0.650061555869014</li>\n",
       "</ol>\n"
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       "\\begin{enumerate*}\n",
       "\\item 3.28678678551627\n",
       "\\item 0.31387097276445\n",
       "\\item -3.02500161149644\n",
       "\\item 1.36130631294013\n",
       "\\item 0.650061555869014\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 3.28678678551627\n",
       "2. 0.31387097276445\n",
       "3. -3.02500161149644\n",
       "4. 1.36130631294013\n",
       "5. 0.650061555869014\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  3.2867868  0.3138710 -3.0250016  1.3613063  0.6500616"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "<table>\n",
       "<tbody>\n",
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       "\\begin{tabular}{l}\n",
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       "\t 1\\\\\n",
       "\t 1\\\\\n",
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       "1. 1\n",
       "2. 1\n",
       "3. 1\n",
       "4. 1\n",
       "5. 1\n",
       "\n",
       "\n"
      ],
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       "     [,1]\n",
       "[1,]    1\n",
       "[2,]    1\n",
       "[3,]    1\n",
       "[4,]    1\n",
       "[5,]    1"
      ]
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     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# solve a linear system using R\n",
    "A <- array(runif(25), dim=c(5,5)) # 5 by 5 matrix with uniformly distributed elements\n",
    "b <- rep(1,5);\n",
    "x <- solve(A,b);\n",
    "x\n",
    "A %*% x   # check that x solves the linear system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 4.992164</td><td> 2.381509</td><td> 5.862320</td><td>-6.310363</td><td>-3.638844</td></tr>\n",
       "\t<tr><td> 0.282778</td><td>-1.400723</td><td>-1.985402</td><td> 1.730754</td><td> 1.686464</td></tr>\n",
       "\t<tr><td>-4.7926526</td><td>-1.1627247</td><td>-2.8455445</td><td> 4.9559795</td><td> 0.8199406</td></tr>\n",
       "\t<tr><td> 0.7890497</td><td>-0.3361745</td><td> 0.2298502</td><td>-0.7290643</td><td> 1.4076452</td></tr>\n",
       "\t<tr><td>-0.008949328</td><td> 1.184794406</td><td>-0.242017138</td><td> 0.032201496</td><td>-0.315967880</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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      "text/latex": [
       "\\begin{tabular}{lllll}\n",
       "\t  4.992164 &  2.381509 &  5.862320 & -6.310363 & -3.638844\\\\\n",
       "\t  0.282778 & -1.400723 & -1.985402 &  1.730754 &  1.686464\\\\\n",
       "\t -4.7926526 & -1.1627247 & -2.8455445 &  4.9559795 &  0.8199406\\\\\n",
       "\t  0.7890497 & -0.3361745 &  0.2298502 & -0.7290643 &  1.4076452\\\\\n",
       "\t -0.008949328 &  1.184794406 & -0.242017138 &  0.032201496 & -0.315967880\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 4.99216436832737\n",
       "2. 0.282778032083129\n",
       "3. -4.79265255000991\n",
       "4. 0.789049684877187\n",
       "5. -0.0089493276662397\n",
       "6. 2.38150925818533\n",
       "7. -1.40072320883759\n",
       "8. -1.16272472194479\n",
       "9. -0.33617445239323\n",
       "10. 1.18479440581685\n",
       "11. 5.86232032633008\n",
       "12. -1.98540195904193\n",
       "13. -2.84554449389478\n",
       "14. 0.229850205411154\n",
       "15. -0.242017138489628\n",
       "16. -6.31036328829249\n",
       "17. 1.73075411896867\n",
       "18. 4.95597950815158\n",
       "19. -0.729064313943304\n",
       "20. 0.0322014960323736\n",
       "21. -3.63884387903403\n",
       "22. 1.68646398959218\n",
       "23. 0.81994064620145\n",
       "24. 1.40764518898833\n",
       "25. -0.31596787982434\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "             [,1]       [,2]       [,3]       [,4]       [,5]\n",
       "[1,]  4.992164368  2.3815093  5.8623203 -6.3103633 -3.6388439\n",
       "[2,]  0.282778032 -1.4007232 -1.9854020  1.7307541  1.6864640\n",
       "[3,] -4.792652550 -1.1627247 -2.8455445  4.9559795  0.8199406\n",
       "[4,]  0.789049685 -0.3361745  0.2298502 -0.7290643  1.4076452\n",
       "[5,] -0.008949328  1.1847944 -0.2420171  0.0322015 -0.3159679"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
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       "\t<tr><td>-3.025002</td></tr>\n",
       "\t<tr><td>1.361306</td></tr>\n",
       "\t<tr><td>0.6500616</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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      "text/latex": [
       "\\begin{tabular}{l}\n",
       "\t 3.286787\\\\\n",
       "\t 0.313871\\\\\n",
       "\t -3.025002\\\\\n",
       "\t 1.361306\\\\\n",
       "\t 0.6500616\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "1. 3.28678678551627\n",
       "2. 0.31387097276445\n",
       "3. -3.02500161149644\n",
       "4. 1.36130631294013\n",
       "5. 0.650061555869014\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "           [,1]\n",
       "[1,]  3.2867868\n",
       "[2,]  0.3138710\n",
       "[3,] -3.0250016\n",
       "[4,]  1.3613063\n",
       "[5,]  0.6500616"
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     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
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    {
     "data": {
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       "\\begin{tabular}{l}\n",
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      "text/markdown": [
       "1. 1\n",
       "2. 1\n",
       "3. 0.999999999999999\n",
       "4. 0.999999999999999\n",
       "5. 1\n",
       "\n",
       "\n"
      ],
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       "     [,1]\n",
       "[1,]    1\n",
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     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "Ainv = solve(A);\n",
    "Ainv\n",
    "x <- Ainv %*% b;\n",
    "x\n",
    "A %*% x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next chapter: [Probability,Random Vectors, and the Multivariate Normal Distribution](prob.ipynb)"
   ]
  }
 ],
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