Complex Matrices.
What is the Hermitian transpose of matrix, and how is this different from the standard transpose? What property does a Hermitian matrix possess? Prove that the diagonal entries of a Hermitian matrix must be real. What property does a unitary matrix possess? What can you say about the eigenvalues and eigenvectors of a unitary matrix?
Hermitian Square of a Matrix and the Matrix Square Root. What properties does a Hermitian positive semi-definite matrix possess? Given any matrix A, prove that the matrix AHA is Hermitian positive semi-definite.