Show that each matrix type is normal.
I will use the notation:
A^T = transpose(A)
A^* = complex_conjugate(A)
A^t = (A^T)^* = hermitian_conjugate(A)
A^i = inverse(A)
A matrix, A, is defined to normal if and only if
[A , A^t] = A A^t - A^t A = 0
(comment: A matrix, A, is also normal if and only if there exists a unitary matrix, U, such that (U A U^t) is ...
This shows how to prove that each matrix type is normal.