Normal Matricies?
How do I show that a matrix A in Mn is normal IFF it commutes with some normal matrix with distinct eigenvalues?
A square matrix A 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 m can be tested to see if it is normal using the Mathematica function
NormalQ[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., 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, ….
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