Editing Diagonal matrix (section)

== Applications ==
Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is typically desirable to represent a given matrix or [[linear operator|linear map]] by a diagonal matrix.

In fact, a given {{mvar|n}}-by-{{mvar|n}} matrix {{math|'''A'''}} is [[similar matrix|similar]] to a diagonal matrix (meaning that there is a matrix {{math|'''X'''}} such that {{math|'''X'''<sup>−1</sup>'''AX'''}} is diagonal) if and only if it has {{mvar|n}} [[linearly independent]] eigenvectors. Such matrices are said to be [[diagonalizable matrix|diagonalizable]].

Over the [[field (mathematics)|field]] of [[real number|real]] or [[complex number|complex]] numbers, more is true. The [[spectral theorem]] says that every [[normal matrix]] is [[matrix similarity|unitarily similar]] to a diagonal matrix (if {{math|1='''AA'''<sup>∗</sup> = '''A'''<sup>∗</sup>'''A'''}} then there exists a [[unitary matrix]] {{math|'''U'''}} such that {{math|'''UAU'''<sup>∗</sup>}} is diagonal). Furthermore, the [[singular value decomposition]] implies that for any matrix {{math|'''A'''}}, there exist unitary matrices {{math|'''U'''}} and {{math|'''V'''}} such that {{math|'''U'''<sup>∗</sup>'''AV'''}} is diagonal with positive entries.