What does it mean to say an matrix is Hermitian?
What does it mean to say an matrix is unitary?
Show that the eigenvalues of a Hermitian matrix are real and that eigenvectors corresponding to distinct eigenvalues are orthogonal.
Suppose that is an Hermitian matrix with distinct eigenvalues and corresponding normalised eigenvectors . Let denote the matrix whose columns are . Show directly that is unitary and , where is a diagonal matrix you should specify.
If is unitary and diagonal, must it be the case that is Hermitian? Give a proof or counterexample.
Find a unitary matrix and a diagonal matrix such that