A complex square matrix A is a normal matrix iff

$A^\{*\}A=AA^\{*\}$

where A^* is the conjugate transpose of A (if A is a real matrix, this is the same as the transpose of A).

Examples

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All unitary, hermitian, and skew-hermitian matrices are normal. If A is unitary A^*A=AA^*=I. If A is hermitian, then A^*=A and so AA^*=AA=A^*A.

But it is not the case that all normal matrices are either unitary or (skew-)hermitian; for example,

$\backslash begin\{pmatrix\}$

-i & -i \\ -i & i \end{pmatrix} is normal since

$\backslash begin\{pmatrix\}$

-i & -i \\ -i & i \end{pmatrix}\begin{pmatrix} -i & -i \\ -i & i \end{pmatrix}^* = \begin{pmatrix} -i & -i \\ -i & i \end{pmatrix}\begin{pmatrix} i & i \\ i & -i \end{pmatrix}

$=\; \backslash begin\{pmatrix\}$

2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} i & i \\ i & -i \end{pmatrix}\begin{pmatrix} -i & -i \\ -i & i \end{pmatrix} = \begin{pmatrix} -i & -i \\ -i & i \end{pmatrix}^*\begin{pmatrix} -i & -i \\ -i & i \end{pmatrix} but the matrix is clearly not unitary, nor is it hermitian.

Consequences

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It is useful to think of normal matrices in analogy to complex numbers, invertible normal matrices in analogy to non-zero complex numbers, the conjugate transpose in analogy to the complex conjugate, unitary matrices in analogy to complex numbers of absolute value 1, hermitian matrices in analogy to real numbers and hermitian positive definite matrices in analogy to positive real numbers.

The concept of normality is mainly important because normal matrices are precisely the ones to which the spectral theorem applies; in other words, normal matrices are precisely those matrices that can be represented by a diagonal matrix with respect to a properly chosen orthonormal basis of Cⁿ. Phrased differently: a matrix is normal if and only if its eigenspaces span Cⁿ and are pairwise orthogonal with respect to the standard inner product of Cⁿ.

In general, the sum or product of two normal matrices need not be normal. However, if A and B are normal with AB = BA, then both AB and A + B are also normal and furthermore we can simultaneously diagonalize A and B in the following sense: there exists a unitary matrix U such UAU^* and UBU^* are both diagonal matrices. In this case, the columns of U^* are eigenvectors of both A and B and form an orthonormal basis of Cⁿ.

If A is both a triangular matrix and a normal matrix, then A is diagonal. This can be seen by looking at the diagonal entries of A^*A and AA^*, where A is a normal, triangular matrix.

If A is an invertible normal matrix, then there exists a unitary matrix U and a hermitian positive definite matrix R such that A = RU = UR. The matrices R and U are uniquely determined by A. This statement can be seen as an analog (and generalization) of the polar representation of non-zero complex numbers.

The concept of normal matrices can be generalized to normal operators on Hilbert spaces and to normal elements in C-star algebras.

Matrices

Normale Matrix | Matrice normale | Matrice normale | טרנספורמציה נורמלית | Нормальная матрица