Page 326 - A Course in Linear Algebra with Applications

3 1 0 Chapter Nine: Advanced Topics

where U is the unitary matrix

^ 2 V 0 0 V2/

Normal matrices
We have seen that every nxn hermitian matrix A has the
n
property that there is an orthonormal basis of C consisting
of eigenvectors of A. It was also observed that this property
immediately leads to A being diagonalizable by a unitary ma-
trix, namely the matrix whose columns are the vectors of the
orthonormal basis. We shall consider what other matrices
have this useful property.
A complex matrix A is called normal if it commutes with
its complex transpose,

A* A = AA*.

Of course for a real matrix this says that A commutes with
T
its transpose A . Clearly hermitian matrices are normal; for
if A = A*, then certainly A commutes with A*. What is the
connection between normal matrices and the existence of an
orthonormal basis of eigenvectors? The somewhat surprising
answer is given by the next theorem.

Theorem 9.1.5
Let A be a complex nxn matrix. Then A is normal if and
only if there is an orthonormal basis of C n consisting of eigen-
vectors of A.

Proof
First of all suppose that C n has an orthonormal basis of
eigenvectors of A. Then, as has been noted, there is a uni-
tary matrix U such that U*AU = D is diagonal. This leads