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In fact, a given ''n''-by-''n'' matrix ''A'' is [[similar matrix|similar]] to a diagonal matrix (meaning that there is a matrix ''X'' such that ''X''<sup>−1</sup>''AX'' is diagonal) if and only if it has ''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 ''AA''<sup>∗</sup> = ''A''<sup>∗</sup>''A'' then there exists a [[unitary matrix]] ''U'' such that ''UAU''<sup>∗</sup> is diagonal). Furthermore, the [[singular value decomposition]] implies that for any matrix ''A'', there exist unitary matrices ''U'' and ''V'' such that ''
== Operator theory ==
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