If $A$ is symmetrical, then $B^TAB$ is symmetrical for every matrix $B$ for which the product is defined.

Question

How do I show that if $A$ is a symmetrical matrix, then $B^TAB$ is symmetrical for every matrix $B$ for which a product is defined?

I don't know how to show that the statement is true in the general case. If $A$ is symmetrical then that means that is transpose is identical to itself. That means you could write the original $B^TAB$ into $(AB)^TB$. But I don't know why I would want to do that.

Answer 1

$$(B^TAB)^T=B^TA^T(B^T)^T=B^TAB$$