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eigenvalues of a Hermitian matrix are real

(Theorem)

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

Proof. Suppose $\lambda$ is an eigenvalue of the self-adjoint matrix

with non-zero eigenvector

. Then $Av = \lambda v$ .

$\displaystyle \lambda ^{\ast }v^{H}v=\left( \lambda v\right) ^{H}v=\left( Av\right) ^{H}v=v^{H}A^{H}v=v^{H}Av=v^{H}\lambda v=\lambda v^{H}v$

Since is non-zero by assumption, is non-zero as well and so $\lambda^{*}=\lambda$ , meaning that $\lambda$ is real. $\qedsymbol$

"eigenvalues of a Hermitian matrix are real" is owned by Andrea Ambrosio. [ full author list (2) | owner history (1) ]

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Cross-references: eigenvector, real, matrix, Hermitian, eigenvalues
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This is version 5 of eigenvalues of a Hermitian matrix are real, born on 2004-06-02, modified 2006-10-10.
Object id is 5879, canonical name is EigenvaluesOfAHermitianMatrixAreReal.
Accessed 7613 times total.

Classification:

15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )

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