corollary of Schur decomposition

theorem:ACn×n is a normal matrixMathworldPlanetmath if and only if there exists a unitary matrixMathworldPlanetmath Qn×n such that QHAQ=diag(λ1,,λn)(the diagonal matrixMathworldPlanetmath) where H is the conjugate transposeMathworldPlanetmath. [GVL]

proof: Firstly we show that if there exists a unitary matrix Qn×n such that QHAQ=diag(λ1,,λn) then An×n is a normal matrix. Let D=diag(λ1,,λn) then A may be written as A=QDQH. Verifying that A is normal follows by the following observation AAH=QDQHQDHQH=QDDHQH and AHA=QDHQHQDQH=QDHDQH. Therefore A is normal matrix because DDH=diag(λ1λ1¯,,λnλn¯)=DHD.
Secondly we show that if An×n is a normal matrix then there exists a unitary matrix Qn×n such that QHAQ=diag(λ1,,λn). By Schur decompostion we know that there exists a Qn×n such that QHAQ=T(T is an upper triangular matrixMathworldPlanetmath). 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).


  • 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