# Difference between revisions of "Normal matrices"

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− | *[http://en.wikipedia.org/wiki/ | + | *[http://en.wikipedia.org/wiki/Normal_matrix Normal_matrix entry in Wikipedia] |

## 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 .