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:
- A is normal.
- A is diagonalizable by a unitary matrix.
- The entire space is spanned by some orthonormal set of eigenvectors of A.
- for every x.
- (That is, the Frobenius norm of A can be computed by the eigenvalues of A.)
- The Hermitian part and skew-Hermitian part of A commute.
- is a polynomial (of degree ≤ n − 1) in .
- for some unitary matrix U.
- U and P commute, where we have the polar decomposition A = UP with a unitary matrix U and some positive semidefinite matrix P.
- A commutes with some normal matrix N with distinct eigenvalues.
- 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:
Read more about this topic: Normal Matrix
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