diff --git a/lineax/_operator.py b/lineax/_operator.py
index d8ab745..f5f3663 100644
--- a/lineax/_operator.py
+++ b/lineax/_operator.py
@@ -1514,7 +1514,11 @@ def is_diagonal(operator: AbstractLinearOperator) -> bool:
 @is_diagonal.register(JacobianLinearOperator)
 @is_diagonal.register(FunctionLinearOperator)
 def _(operator):
-    return diagonal_tag in operator.tags
+    return (
+        diagonal_tag in operator.tags
+        or operator.in_size() == 1
+        and operator.out_size() == 1
+    )
 
 
 @is_diagonal.register(IdentityLinearOperator)
@@ -1525,7 +1529,7 @@ def _(operator):
 
 @is_diagonal.register(TridiagonalLinearOperator)
 def _(operator):
-    return False
+    return operator.in_size() == 1
 
 
 # is_tridiagonal