Some Results for Hermitian Matrix

Lemma 1: All eigenvalues of a Hermitian matrix are real. Proof: Let be Hermitian and and be an eigenvalue and the corresponding eigenvector of . We have . Thus we have as is real in general. Lemma 2: Two eigenvectors of a Hermitian matrix are orthogonal to each other if their eigenvalues are different. Proof:…

Schur Complement and Positive Definite Matrix

For a matrix , we call the Schur complement of in . Note that naturally appear in block matrix inversion. Note that when is symmetric and is positive definite, is positive definite if and only if is also positive definite. The proof is rather straightforward. Consider the function . Let’s try to minimize w.r.t. ,…