In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*:
A normal ⟺ A ∗ A = A 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; implying in particular that 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 A = U D U ∗ {\displaystyle A=UDU^{*}} and A ∗ = U D ∗ U ∗ {\displaystyle A^{*}=UD^{*}U^{*}} where D {\displaystyle D} is a diagonal matrix whose diagonal values are in general complex and U {\displaystyle U} is a unitary matrix. The left and right singular vectors in the singular value decomposition of a normal matrix A = U D V ∗ {\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.
Special cases Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal, with all eigenvalues being unit modulus, real, and imaginary, respectively. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal, with all eigenvalues being complex conjugate pairs on the unit circle, real, and imaginary, respectively. However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian, as their eigenvalues can be any complex number, in general. For example,
A = [ 1 1 0 0 1 1 1 0 1 ] {\displaystyle A={\begin{bmatrix}1&1&0\\0&1&1\\1&0&1\end{bmatrix}}}
is neither unitary, Hermitian, nor skew-Hermitian, because its eigenvalues are 2 , ( 1 ± i 3 ) / 2 {\displaystyle 2,(1\pm i{\sqrt {3}})/2} ; yet it is normal because
A A ∗ = [ 2 1 1 1 2 1 1 1 2 ] = A ∗ A . {\displaystyle AA^{*}={\begin{bmatrix}2&1&1\\1&2&1\\1&1&2\end{bmatrix}}=A^{*}A.}
Consequences
The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies:
The diagonal entries of Λ are the eigenvalues of A, and the columns of U are the eigenvectors of A. The matching eigenvalues in Λ come in the same order as the eigenvectors are ordered as columns of U. Another way of stating the spectral theorem is to say that 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. The spectral theorem for normal matrices is a special case of the more general Schur decomposition which holds for all square matrices. Let A be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. If A is normal, so is B. But then B must be diagonal, for, as noted above, a normal upper-triangular matrix is diagonal. The spectral theorem permits the classification of normal matrices in terms of their spectra, for example:
In general, the sum or product of two normal matrices need not be normal. However, the following holds:
In this special case, the columns of U* are eigenvectors of both A and B and form an orthonormal basis in Cn. This follows by combining the theorems that, over an algebraically closed field, commuting matrices are simultaneously triangularizable and a normal matrix is diagonalizable – the added result is that these can both be done simultaneously.
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