Normal Matrix - Equivalent Definitions

Equivalent Definitions

It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let A be a n-by-n complex matrix. Then the following are equivalent:

1. A is normal.
2. A is diagonalizable by a unitary matrix.
3. The entire space is spanned by some orthonormal set of eigenvectors of A.
4. for every x.
5. (That is, the Frobenius norm of A can be computed by the eigenvalues of A.)
6. The Hermitian part and skew-Hermitian part of A commute.
7. is a polynomial (of degree ≤ n − 1) in .
8. for some unitary matrix U.
9. U and P commute, where we have the polar decomposition A = UP with a unitary matrix U and some positive semidefinite matrix P.
10. A commutes with some normal matrix N with distinct eigenvalues.
11. for all where A has singular values and eigenvalues

Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces. For example, a bounded operator satisfying (9) is only quasinormal.

The operator norm of a normal matrix N equals the spectral and numerical radii of N. (This fact generalizes to normal operators.) Explicitly, this means: