Normal matrices: Difference between revisions
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(New page: A complex square matrix A is a normal matrix if :<math>A^\dagger A=AA^\dagger ,</math> where <math>A^\dagger</math> is the conjugate transpose of A. That is, a matrix is normal if it [[c...) |
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Revision as of 12:08, 11 February 2008
A complex square matrix A is a normal matrix if
where is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose: .
Normal matrices are precisely those to which the spectral theorem applies: a matrix is normal if and only if it can be represented by a diagonal matrix and a unitary matrix by the formula
where
The entries of the diagonal matrix are the eigenvalues of , and the columns of are the eigenvectors of . The matching eigenvalues in must be ordered as the eigenvectors are ordered as columns of .