As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting.
(The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.)
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.
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.
Proposition — A normal triangular matrix is diagonal.
Let A be any normal upper triangular matrix.
using subscript notation, one can write the equivalent expression using instead the ith unit vector (
This implies the first row must be zero for entries 2 through n. Continuing this argument for row–column pairs 2 through n shows A is diagonal.
The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies: Proposition — A matrix A is normal if and only if there exists a diagonal matrix Λ and a unitary matrix U such that A = UΛU*.
The diagonal entries of Λ are the eigenvalues of A, and the columns of U are the eigenvectors of A.
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.
Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B.
The spectral theorem permits the classification of normal matrices in terms of their spectra, for example: Proposition — A normal matrix is unitary if and only if all of its eigenvalues (its spectrum) lie on the unit circle of the complex plane.
Proposition — A normal matrix is self-adjoint if and only if its spectrum is contained in
In other words: A normal matrix is Hermitian if and only if all its eigenvalues are real.
Furthermore there exists a unitary matrix U such that UAU* and UBU* are diagonal matrices.
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.
It is possible to give a fairly long list of equivalent definitions of a normal matrix.
Then the following are equivalent: Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces.
For example, a bounded operator satisfying (9) is only quasinormal.
It is occasionally useful (but sometimes misleading) to think of the relationships of special kinds of normal matrices as analogous to the relationships of the corresponding type of complex numbers of which their eigenvalues are composed.
This is because any function of a non-defective matrix acts directly on each of its eigenvalues, and the conjugate transpose of its spectral decomposition
Likewise, if two normal matrices commute and are therefore simultaneously diagonalizable, any operation between these matrices also acts on each corresponding pair of eigenvalues.
As a special case, the complex numbers may be embedded in the normal 2×2 real matrices by the mapping
It is easy to check that this embedding respects all of the above analogies.