Normal Matrix

A square matrix A is a normal matrix if


where [a,b] is the commutator and A^(H) denotes the conjugate transpose. For example, the matrix

 [i 0; 0 3-5i]

is a normal matrix, but is not a Hermitian matrix.

A matrix m 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., A is a normal matrix iff there exists a unitary matrix U such that UAU^(-1) 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 n=1, 2, ....

(0,1)A0555472, 8, 68, 1124, ...
(-1,1)A0555482, 12, 80, 2096, ...
(-1,0,1)A0555493, 33, 939, ...

See also

Conjugate Transpose, Diagonal Matrix, Diagonalizable Matrix, Hermitian Matrix, Normal Equation, Unitary Matrix

Explore with Wolfram|Alpha


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

Referenced on Wolfram|Alpha

Normal Matrix

Cite this as:

Weisstein, Eric W. "Normal Matrix." From MathWorld--A Wolfram Web Resource.

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