Two questions regarding real symmetric matrices.
1) $A$, $B$, $C$, $D$ are $n\times n$ real matrices, $A \sim B$ and $C \sim D$. Then $AC\sim BD$ ($\sim$ = similar).
- I know if $A \sim B$ then $A = PBP^{-1}$. But I believe $AC\sim BD$ is not true.
2) If $A$ and $B$ are $n\times n$ real symmetric matrices and congruent then they are also similar.
- I know if they are congruent then there exists an invertible matrix $P$ such that $P^TBP = A$ and they have the same rank. I see that $P^T = P^{-1}$ then it is orthogonal also. But not sure if "similar".