My understanding is that the multiplication of two matrices is NOT commutative most of the time.
One exception is two matrices, A and B, that are inverses of the other. This condition leads in turn, to some important restrictions.
- The two matrices are square matrices of the same dimension.
- One equals the transpose of the other, divided by their (common) determinant, to "unitize" the inverse matrix.
- They are "nonsingular" insofar as their determinant is non-zero (can't divide by zero in 2, above).
What causes such matrices to be commutative under multiplication? Is invertibility a necessary and/or sufficient condition for two matrices to be commutative in this way?
In this post, there was an answer that the required condition was "a common basis of generalized eigenvectors." How does that allow commutativity between the two matrices? Is is because they are "bijective" (injective and surjective)?