A complex square matrix A is a normal matrix if In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

A*A=AA*

where A* is the conjugate transpose of A. (If A is a real matrix, A*=A^T and so it is normal if A^TA = AA^T.) In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Examples

All unitary, Hermitian, and skew-Hermitian matrices are normal. (For real matrices, this simplifies as orthogonal, symmetric, and skew-symmetric matrices are normal.) If A is unitary A*A = AA* = I. If A is Hermitian, then A* = A and so AA* = AA = A*A. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ...

However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian. As an example, the matrix

is normal because

.

The matrix A is neither unitary, Hermitian, nor skew-Hermitian.

Consequences

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. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...

The concept of normality is important (especially in the more general, infinite dimensional setting) 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 mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

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. In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...

Any square matrix A has polar decomposition A = UP where U is unitary and P positive semidefinite. (If A is invertible, then U and P are uniquely determined by A.) If A is normal, then UP = PU. The converse is true only in the finite dimensional case. In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to polar decomposition of a nonzero complex number z where r is the absolute value of z (a positive real number), and is the complex sign of z. ...

The concept of normal matrices can be extended to normal operators on infinite dimensional Hilbert 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. In functional analysis, a normal operator on a Hilbert space is a continuous linear operator that commutes with its hermitian adjoint : The main importance of this concept is that the spectral theorem applies to normal operators. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... C*-algebras are an important area of research in functional analysis. ...