normal matrix
A complex matrix A∈ℂn×n is said to be normal if A∗A=AA∗ where ∗ denotes the conjugate transpose.
Similarly for a real matrix A∈ℝn×n is said to be normal if ATA=AAT where T denotes the transpose.
properties:
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Equivalently a complex matrix A∈ℂn×n is said to be normal if it satisfies [A,A∗]=0 where [,] is the commutator bracket.
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Equivalently a real matrix A∈ℝn×n is said to be normal if it satisfies [A,AT]=0 where [,] is the commutator bracket.
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Let A be a square complex matrix of order n. It follows from Schur’s inequality that if A is a normal matrix
then ∑ni=1|λi|2=traceA∗A where ∗ is the conjugate transpose and λi are the eigenvalues
of A.
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A complex square matrix
is diagonal if and only if it is normal, triangular.(see theorem for normal triangular matrices).
examples:
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(ab-ba) where a,b∈ℝ
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(1i-i1)
see also:
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Wikipedia, http://www.wikipedia.org/wiki/Normal_matrixnormal matrix
Title | normal matrix |
---|---|
Canonical name | NormalMatrix |
Date of creation | 2013-03-22 13:41:10 |
Last modified on | 2013-03-22 13:41:10 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 12 |
Author | Daume (40) |
Entry type | Definition |
Classification | msc 15A21 |
Synonym | normal |
Related topic | TheoremForNormalTriangularMatrices |