# Schur decomposition

If $A$ is a complex square matrix^{} of order n (i.e. $A\mathrm{\in}{\mathrm{Mat}}_{n}\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$), then there exists a unitary matrix^{} $Q\in {\mathrm{Mat}}_{n}(\u2102)$ such that

${Q}^{H}AQ=T=D+N$

where ${}^{H}$ is the conjugate transpose^{}, $D=\mathrm{diag}({\lambda}_{1},\mathrm{\dots},{\lambda}_{n})$ (the ${\lambda}_{i}$ are eigenvalues^{} of $A$), and $N\in {\mathrm{Mat}}_{n}(\u2102)$ is strictly upper triangular matrix^{}. Furthermore, $Q$ can be chosen such that the eigenvalues ${\lambda}_{i}$ appear in any order along the diagonal. [GVL]

## References

- GVL Golub, H. Gene, Van Loan F. Charles: Matrix Computations (Third Edition). The Johns Hopkins University Press, London, 1996.

Title | Schur decomposition^{} |
---|---|

Canonical name | SchurDecomposition |

Date of creation | 2013-03-22 13:42:12 |

Last modified on | 2013-03-22 13:42:12 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 8 |

Author | Daume (40) |

Entry type | Theorem |

Classification | msc 15-00 |

Related topic | AnExampleForSchurDecomposition |

Related topic | ProofThatDetEAEoperatornametrA |