normal matrix

It is defined using the Kronecker product and normal matrix addition.

Then the bra can be computed by normal matrix multiplication.

Thus the eigenvalue problem for all normal matrices is well-conditioned.

In general, the sum or product of two normal matrices need not be normal.

Also, if the entries come from the field R or C, then it is a normal matrix as well.

U is a normal matrix with eigenvalues lying on the unit circle.

It is possible to give a fairly long list of equivalent definitions of a normal matrix.

A commutes with some normal matrix N with distinct eigenvalues.

If A is a normal matrix then is the convex hull of its eigenvalues.

Every generalized eigenvector of a normal matrix is an ordinary eigenvector.

As a corollary, we have that every commuting family of normal matrices can be simultaneously diagonalized.

However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian.

Eigenvectors of distinct eigenvalues of a normal matrix are orthogonal.

For example, as mentioned below, the problem of finding eigenvalues for normal matrices is always well-conditioned.

Also, a normal matrix is self-adjoint if and only if its spectrum consists of reals.

A matrix which is simultaneously triangular and normal matrix, is also diagonal.

Every Hermitian matrix is a normal matrix.

In particular, the eigenspace problem for normal matrices is well-conditioned for isolated eigenvalues.

The spectral theorem permits the classification of normal matrices in terms of their spectra.

And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized.

For example, a normal matrix is unitary if and only if its spectrum is contained in the unit circle of the complex plane.

The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix.

This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:

An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.

When comparing the two results, a rough analogy can be made with the relationship between the spectral theorem for normal matrices and the Jordan canonical form.