# corollary of Schur decomposition

theorem:$A\mathrm{\in}{\mathrm{C}}^{n\mathrm{\times}n}$ is a normal matrix^{} if and only if there exists a unitary matrix^{} $Q\in {\u2102}^{n\times n}$ such that ${Q}^{H}AQ=\mathrm{diag}({\lambda}_{1},\mathrm{\dots},{\lambda}_{n})$(the diagonal matrix^{}) where ${}^{H}$ is the conjugate transpose^{}. [GVL]

proof: Firstly we show that if there exists a unitary matrix $Q\in {\u2102}^{n\times n}$ such that ${Q}^{H}AQ=\mathrm{diag}({\lambda}_{1},\mathrm{\dots},{\lambda}_{n})$ then $A\in {\u2102}^{n\times n}$ is a normal matrix. Let $D=\mathrm{diag}({\lambda}_{1},\mathrm{\dots},{\lambda}_{n})$ then $A$ may be written as $A=QD{Q}^{H}$. Verifying that A is normal follows by the following observation $A{A}^{H}=QD{Q}^{H}Q{D}^{H}{Q}^{H}=QD{D}^{H}{Q}^{H}$ and ${A}^{H}A=Q{D}^{H}{Q}^{H}QD{Q}^{H}=Q{D}^{H}D{Q}^{H}$. Therefore $A$ is normal matrix because $D{D}^{H}=\mathrm{diag}({\lambda}_{1}\overline{{\lambda}_{1}},\mathrm{\dots},{\lambda}_{n}\overline{{\lambda}_{n}})={D}^{H}D$.

Secondly we show that if $A\in {\u2102}^{n\times n}$ is a normal matrix then there exists a unitary matrix $Q\in {\u2102}^{n\times n}$ such that ${Q}^{H}AQ=\mathrm{diag}({\lambda}_{1},\mathrm{\dots},{\lambda}_{n})$. By Schur decompostion we know that there exists a $Q\in {\u2102}^{n\times n}$ such that ${Q}^{H}AQ=T$($T$ is an upper triangular matrix^{}). Since $A$ is a normal matrix then $T$ is also a normal matrix. The result that $T$ is a diagonal matrix comes from showing that a normal upper triangular matrix is diagonal (see theorem for normal triangular matrices).

QED

## References

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

Title | corollary of Schur decomposition |
---|---|

Canonical name | CorollaryOfSchurDecomposition |

Date of creation | 2013-03-22 13:43:38 |

Last modified on | 2013-03-22 13:43:38 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 7 |

Author | Daume (40) |

Entry type | Corollary |

Classification | msc 15-00 |