Congruence transformation of symmetric matrices

Question

Given a symmetric matrix $A$ of size $n$ and an arbitrary invertible square matrix $P$ (also of size $n$), what can we say about the congruence transformation: $$P^TAP?$$

• Does this matrix necessarily have to be a symmetric matrix?
• When is the corresponding matrix symmetric (and not symmetric)?

Answer 1

You have $$ (P^TAP)^T=P^TA^TP^{TT}=P^TAP. $$