# matrix monotone

A real function $f$ on a real interval $I$ is said to be *matrix monotone of $n$*, if

$$A{\le}_{L}B\Rightarrow f(A)\le f(B)$$ | (1) |

for all Hermitian $n\times n$ matrices $A,B$ with spectra contained in $I$. Here ${\le}_{L}$ denotes the Loewner order, and the notation $f(A)$ is explained in the entry functional calculus for Hermitian matrices.

Title | matrix monotone |
---|---|

Canonical name | MatrixMonotone |

Date of creation | 2013-03-22 13:34:57 |

Last modified on | 2013-03-22 13:34:57 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 12 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 40A30 |

Related topic | ALeqBForHermitianMatricesAB |

Related topic | OperatorMonotone |