# commuting matrices are simultaneously triangularizable

###### Theorem 1.

All matrices in the below are complex $n\mathrm{\times}n$ matrices.

Let $A$,$B$ be matrices and $A\mathit{}B\mathrm{=}B\mathit{}A$. Then there exists a unitary matrix^{} $Q$ such that

${Q}^{H}AQ={T}_{1}$ , ${Q}^{H}\mathit{}B\mathit{}Q\mathrm{=}{T}_{\mathrm{2}}$

where ${}^{H}$ is the conjugate transpose^{} and ${T}_{\mathrm{1}}\mathrm{,}{T}_{\mathrm{2}}\mathrm{,}$ are upper triangular matrices^{}.

Title | commuting matrices are simultaneously triangularizable |
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Canonical name | CommutingMatricesAreSimultaneouslyTriangularizable |

Date of creation | 2013-03-22 15:26:48 |

Last modified on | 2013-03-22 15:26:48 |

Owner | georgiosl (7242) |

Last modified by | georgiosl (7242) |

Numerical id | 12 |

Author | georgiosl (7242) |

Entry type | Theorem |

Classification | msc 15A23 |

Related topic | SimultaneousUpperTriangularBlockDiagonalizationOfCommutingMatrices |