EigenvaluesOfAHermitianMatrixAreReal 2004-06-02 13:43:02 2006-10-10 02:16:49 Theorem Hermitian matrix \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} The eigenvalues of a Hermitian (or self-adjoint) matrix are real. \begin{proof} Suppose $\lambda$ is an eigenvalue of the self-adjoint matrix $A$ with non-zero eigenvector $v$. Then $Av = \lambda v$. \[ \lambda ^{\ast }v^{H}v=\left( \lambda v\right) ^{H}v=\left( Av\right) ^{H}v=v^{H}A^{H}v=v^{H}Av=v^{H}\lambda v=\lambda v^{H}v \] Since $v$ is non-zero by assumption, $v^H v$ is non-zero as well and so $\lambda^{*}=\lambda$, meaning that $\lambda$ is real. \end{proof}