A complex matrix A is called normal if it commutes with its Hermitian transpose: A†A = AAT.
(a) Show that every real symmetric matrix is normal.
(b) Show that every unitary matrix is normal.
(c) Show that every real orthogonal matrix is normal.
(d) Show that an upper triangular matrix is normal if and only if it is diagonal.
(e) Show that the eigenvectors of a normal matrix form an orthogonal basis of Cn under the Hermitian dot product.
(f) Show that the converse is true: a matrix has an orthogonal eigenvector basis of Cn if and only if it is normal.
(g) How can you tell when a real matrix has a real orthonormal eigenvector basis?