Editing Normal matrix (section)

{{Short description|Matrix that commutes with its conjugate transpose}}
In mathematics, a [[complex number|complex]] [[square matrix]] {{mvar|A}} is '''normal''' if it [[commute (mathematics)|commute]]s with its [[conjugate transpose]] {{math|''A''{{sup|*}}}}:
:<math>A \text{ normal} \iff A^*A = AA^* .</math>

The concept of normal matrices can be extended to [[normal operator]]s on [[dimension (vector space)|infinite-dimensional]] [[normed space]]s and to normal elements in [[C*-algebra]]s. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.

The [[spectral theorem]] states that a matrix is normal if and only if it is [[similar matrix|unitarily similar]] to a [[diagonal matrix]], and therefore any matrix {{mvar|A}} satisfying the equation {{math|1=''A''<sup>*</sup>''A'' = ''AA''<sup>*</sup>}} is [[Diagonalizable matrix | diagonalizable]].  (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus <math>A = U D U^*</math> and <math>A^* = U D^* U^*</math>where <math>D</math> is a diagonal matrix whose diagonal values are in general complex.

The left and right singular vectors in the [[singular value decomposition]] of a normal matrix <math>A = U D V^*</math> differ only in complex phase from each other and from the corresponding eigenvectors, since the phase must be factored out of the eigenvalues to form singular values.