Orthogonally equivalent matrices

Question

Suppose $A$ is any square matrix, then how do I show that there exists two orthogonal matrices $Q$ and $R$ such that $Q^TAR=A^T$?

I can show this when $A$ has distinct eigenvalues, but how do I show this for any square matrix $A$? I would really like someone to explain to me how $Q$ and $R$ act on the matrix.

In general, if $A$ and $B$ are two square matrices with the same eigenvalues, then when do have existence of two orthogonal matrices $Q_1$ and $Q_2$ such that $Q_1^TAQ_2=B$?

Let's assume the matrices to be real. Also if you have any reference materials on this, please share.

Thanks for the help.

Answer 1

Let $A=PU$ be a polar decomposition. Then $A^T=U^TP=U^TAU^T$. (Note that this is $U^TAU^T$, not $U^TAU$ or $UAU^T$. So, this is not a similarity transform.)