A square matrix 
is a normal matrix if
![[A,A^(H)]=AA^(H)-A^(H)A=0,](/images/equations/NormalMatrix/NumberedEquation1.svg) |
where ![[a,b]](/images/equations/NormalMatrix/Inline2.svg)
is the commutator and 
denotes the conjugate
transpose. For example, the matrix
![[i 0; 0 3-5i]](/images/equations/NormalMatrix/NumberedEquation2.svg) |
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, ....
