Proposition a matrix a hnn is hermit ian positive definite

Question

Question: Prove Proposition.

Proposition: A matrix A ∈ H^n×n is hermit Ian, positive definite, positive semi definite, skewhermitian, unitary, or normal if and only if χ(A) is symmetric, positive definite, positive semi definite, skew symmetric, orthogonal, or normal, respectively.