A complex matrix H is called Hermitian if it equals its Hermitian adjoint, W = H, as

Question:

A complex matrix H is called Hermitian if it equals its Hermitian adjoint, W = H, as defined in the preceding exercise.
(a) Prove that the diagonal entries of a Hermitian matrix are real.
(b) Prove that (Hz) ∙ w = z ∙ (H w) forz, w ∈ C".
(c) Prove that every Hermitian inner product on C" has the form (z, w) = zT H w where H is an n x n positive definite Hermitian matrix.
(d) How would you verify positive definiteness of a complex matrix?