Normal matrix

In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A^*:

${\displaystyle A{\text{ normal}}\iff A^{*}A=AA^{*}.}$

The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. 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 unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A^*A = AA^* is diagonalizable. (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus ${\displaystyle A=UDU^{*}}$ and ${\displaystyle A^{*}=UD^{*}U^{*}}$where ${\displaystyle D}$ 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 ${\displaystyle A=UDV^{*}}$ 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.