Day: October 21, 2012

Lemma 1: All eigenvalues of a Hermitian matrix are real. Proof: Let $latex A$ be Hermitian and $latex \lambda$ and $latex x$ be an eigenvalue and the corresponding eigenvector of $latex A$. We have $latex \lambda^* x^H x = (x^H A^H) x = x^H (A x)=\lambda x^Hx$. Thus we have $latex \lambda^*=\lambda$ as $latex x^H…

For a matrix $latex M = \begin{pmatrix} A &B\\C&D\end{pmatrix}$, we call $latex S\triangleq D -CA^{-1}B$ the Schur complement of $latex A$ in $latex M$. Note that $latex S$ naturally appear in block matrix inversion. Note that when $latex M$ is symmetric and $latex A$ is positive definite, $latex M$ is positive definite if and only…