# 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.
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Title | eigenvalues of a Hermitian matrix are real |
---|---|

Canonical name | EigenvaluesOfAHermitianMatrixAreReal |

Date of creation | 2013-03-22 14:23:09 |

Last modified on | 2013-03-22 14:23:09 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 8 |

Author | Andrea Ambrosio (7332) |

Entry type | Theorem |

Classification | msc 15A57 |