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# Normal Matrix

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, ....

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

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

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## References

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

Normal Matrix

## Cite this as:

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