# functional calculus for Hermitian matrices

Let $I\subset \mathbb{R}$ be a real interval, $f$ a real-valued function on $I$, and let $M$ be an $n\times n$ real symmetric^{} (and thus Hermitian) matrix whose eigenvalues^{} are contained in $I$.

By the spectral theorem^{}, we can diagonalize $M$ by an orthogonal matrix^{} $O$, so we can write $M=OD{O}^{-1}$ where $D$ is the diagonal matrix^{} consisting of the eigenvalues $\{{\lambda}_{1},{\lambda}_{2},\mathrm{\dots},{\lambda}_{n}\}$. We then define

$f(A)=Of(D){O}^{-1},$ |

where $f(D)$ denotes the diagonal matrix whose diagonal entries are given by $f({\lambda}_{i})$.

It is easy to verify that $f(A)$ is well-defined, i.e. a permutation^{} of the eigenvalues corresponds to the same definition of $f(A)$.

Title | functional calculus for Hermitian matrices |
---|---|

Canonical name | FunctionalCalculusForHermitianMatrices |

Date of creation | 2013-03-22 14:40:12 |

Last modified on | 2013-03-22 14:40:12 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 4 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 47C05 |

Related topic | FunctionalCalculus |