$\begingroup$

Assume we have two commutating matrices, [A,B]=0. Can we say that A and B are symmetric?

Regards

$\endgroup$ 1

1 Answer

$\begingroup$

No, we cannot say in general that $0=[A,B]=AB-BA$ implies that $A$ or $B$ are symmetric: take $A$ non-symmetric and $B$ the identity matrix, for example.
However a related statement is true: if $A$ and $B$ are symmetric, then $AB$ is not symmetric unless $A$ and $B$ commute. Indeed, $(AB)^T=AB$ if and only if $AB=BA$ for symmetric matrices $A,B$, i.e., $[A,B]=0$.

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy