# Wielandt-Hoffman theorem

Let $A$ and $B$ be normal matrices^{} (http://planetmath.org/NormalMatrix). Let their
eigenvalues^{} ${a}_{i}$ and ${b}_{i}$ be ordered such that ${\sum}_{i}{|{a}_{i}-{b}_{i}|}^{2}$ is
minimized. Then we have the following inequality

$$\sum _{i}{|{a}_{i}-{b}_{i}|}^{2}\le {\parallel A-B\parallel}_{F}^{2},$$ |

where $\parallel \cdot {\parallel}_{F}$ is the Frobenius matrix norm.

Title | Wielandt-Hoffman theorem |
---|---|

Canonical name | WielandtHoffmanTheorem |

Date of creation | 2013-03-22 14:58:45 |

Last modified on | 2013-03-22 14:58:45 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 4 |

Author | Andrea Ambrosio (7332) |

Entry type | Theorem |

Classification | msc 15A42 |

Classification | msc 15A18 |

Related topic | ShursInequality |