Some Results for Hermitian Matrix

Lemma 1: All eigenvalues of a Hermitian matrix are real.

Proof: Let A be Hermitian and \lambda and x be an eigenvalue and the corresponding eigenvector of A. We have

\lambda^* x^H x = (x^H A^H) x = x^H (A x)=\lambda x^Hx. Thus we have \lambda^*=\lambda as x^H x is real in general. \Box

Lemma 2: Two eigenvectors of a Hermitian matrix are orthogonal 
to each other if their eigenvalues are different.

Proof: Let x and y be two eigenvectors of a Hermitian matrix A. Let \lambda and \mu be the respective eigenvalues and \lambda \neq \mu. We have

\lambda^*x^Hy=(x^HA^H)y=x^H(Ay)=\mu x^H y. Thus we have (\lambda^* - \mu)x^Hy = (\lambda -\mu)x^Hy=0, where the first equality is from Lemma 1. Since \lambda\neq \mu, we have x^Hy=0. That is, x is orthogonal to y. \Box

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