Diagonalization of an Hermitian matrix

Question

given an Hermitian matrix $$ A = A^{\dagger} $$

is it always true that there wil exist another matrix (unitary) so we have always that

$$ PAP^{\dagger}=D $$ where D is a diagonal matrix

is it true ?

Answer 1

Yes, Hermitian matrix belongs to a class of matrix known as the normal matrix.

Normal matrix is unitarily diagonalizable.

Consider its schur decomposition,

$A=UTU^\dagger$, $A^*=UT^\dagger U^\dagger$, if they are equal, $T$ has to be a diagonal matrix.