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 using the Mathematica function:

  NormalMatrixQ[a_List?MatrixQ] := Module[
    {b = Conjugate @ Transpose @ a},
    a. b === b. a
  ]

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, ....

type	Sloane	counts
	A055547	2, 8, 68, 1124, ...
	A055548	2, 12, 80, 2096, ...
	A055549	3, 33, 939, ...

Sloane, N. J. A. Sequences A055547, A055548, and A055549 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Normal Matrix." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NormalMatrix.html