# eigenvalues of a Hermitian matrix are real

The eigenvalues of a Hermitian (or self-adjoint) matrix are real.

###### 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. ∎

Title eigenvalues of a Hermitian matrix are real EigenvaluesOfAHermitianMatrixAreReal 2013-03-22 14:23:09 2013-03-22 14:23:09 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 8 Andrea Ambrosio (7332) Theorem msc 15A57