A square matrix is a normal matrix if
where is the commutator and denotes the conjugate transpose. For example, the matrix
is a normal matrix, but is not a Hermitian matrix.
A matrix can be tested to see if it is normal in the Wolfram Language using NormalMatrixQ[m].
Normal matrices arise, for example, from a normal equation.
The normal matrices are the matrices which are unitarily diagonalizable, i.e., is a normal matrix iff there exists a unitary matrix such that is a diagonal matrix. All Hermitian matrices are normal but have real eigenvalues, whereas a general normal matrix has no such restriction on its eigenvalues. All normal matrices are diagonalizable, but not all diagonalizable matrices are normal.
The following table gives the number of normal square matrices of given types for orders , 2, ....