Prove that if $AA^T=A^TA$ and $AB=BA$ then $AB^T=B^TA$

Question

Prove that if $AA^T=A^TA$ and $AB=BA$ then $AB^T=B^TA$ where $A$ and $B$ are matrices.

It doesn't say of what order they are. Can somebody help me with this exam problem?

I know I should put my work here but I really don't know what to do. I would really like some help.

What information do you get from $AA^T=A^TA$? Search for matrices that commute with their transpose. They have certain properties. — Ruben Kruepper, Nov 23 '19 at 17:30
Maybe consider $(XY)^T=Y^TX^T$ if I am not mistaken, $AB^T=(BA^T)^T=$.. — Alexey Burdin, Nov 23 '19 at 17:31
Presumably $A$ is real and $B$ is complex. The result is known as Fuglede's theorem and a proof can be found in an earlier thread. — user1551, Nov 23 '19 at 19:22

score 0 · Answer 1 · answered Nov 23 '19 at 19:14

0

Remarks:

1) $AB = B A \Rightarrow B^T A^T = A^T B^T$;

2) $A B^T = B^T A \iff B A^T = A^T B \iff B A^T (AB) = A^T B (AB) $,

answered Nov 23 '19 at 19:14

Marcelo Ng

3

For the last $\iff$ you seem to be assuming $AB$ is invertible. – J.G. Nov 23 '19 at 19:26

1 Answers1