corollary of Schur decomposition
theorem: is a normal matrix if and only if there exists a unitary matrix such that (the diagonal matrix) where is the conjugate transpose. [GVL]
proof: Firstly we show that if there exists a unitary matrix such that then is a normal matrix. Let then may be written as . Verifying that A is normal follows by the following observation and . Therefore is normal matrix because .
Secondly we show that if is a normal matrix then there exists a unitary matrix such that . By Schur decompostion we know that there exists a such that ( is an upper triangular matrix). Since is a normal matrix then is also a normal matrix. The result that 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|
|Date of creation||2013-03-22 13:43:38|
|Last modified on||2013-03-22 13:43:38|
|Last modified by||Daume (40)|