Equivalent definitions It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let be a complex matrix. Then the following are equivalent: is normal. is diagonalizable by a unitary matrix. There exists a set of eigenvectors of which forms an orthonormal basis for . for every . The Frobenius norm of can be computed by the eigenvalues of : . The Hermitian part and skew-Hermitian part of commute. is a polynomial (of degree ) in .Proof: When is normal, use Lagrange's interpolation formula to construct a polynomial such that , where are the eigenvalues of . for some unitary matrix . and commute, where we have the polar decomposition with a unitary matrix and some positive semidefinite matrix . commutes with some normal matrix with distinct eigenvalues. for all where has singular values and has eigenvalues that are indexed with ordering .
equivalent definitions it is possible to give a fairly long list of equivalent definitions of a normal matrix let be a complex matrix then the following are equivalent is normal is diagonalizable by a unitary matrix there exists a set of eigenvectors of which forms an orthonormal basis for  for every  the frobenius norm of can be computed by the eigenvalues of   the hermitian part and skew-hermitian part of commute is a polynomial (of degree ) in proof when is normal use lagrange's interpolation formula to construct a polynomial such that  where are the eigenvalues of  for some unitary matrix  and commute where we have the polar decomposition with a unitary matrix and some positive semidefinite matrix  commutes with some normal matrix with distinct eigenvalues for all where has singular values and has eigenvalues that are indexed with ordering 