If $A$ is positive semidefinite and $P$ is unitary, is $B = P^{-1}AP$ positive semidefinite?

Question

I know that in general, positive semidefiniteness is not preserved by matrix similarity. But is it preserved when $B = P^{-1}AP$ and $P$ is unitary?

Answer 1

In general, symmetry / Hermitianness of matrices is not preserved by a coordinate change. But diagonalizability and all eigenvalues being nonnegative reals (even the exact eigenvalues, including algebraic and geometric multiplicity) is preserved by general coordinate changes. That's because those are properties not of matrices, but of the underlying linear transformations, so coordinate changes do not matter.

A unitary change of coordinates does preserve symmetry / Hermitianness of matrices. So it will also preserve positive semi-definiteness.

Answer 2

Hint: since $P^{-1}=P^*$, $$ \langle x,P^{-1}APx\rangle=\langle Px,APx\rangle $$