Normal Matrix (original) (raw)

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A square matrix is a normal matrix if

[A,A^(H)]=AA^(H)-A^(H)A=0,

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 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 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.