Page 324 - A Course in Linear Algebra with Applications

308 Chapter Nine: Advanced Topics

Proof
By 9.1.3 there is a unitary matrix U such that U* AU = D is
diagonal, with diagonal entries d\,..., d n say. If Xi,..., X n
are the columns of U, then the equation AU = UD implies
that AXi = diXi for i = 1,..., n. Therefore the Xi are eigen-
vectors of A, and since U is unitary, they form an orthonormal
n
basis of C . The argument in the real case is similar.

This justifies our hope that an n x n hermitian matrix
always has enough eigenvectors to form an orthonormal basis
n
of C . Notice that this will be the case even if the eigenvalues
of A are not all distinct.
The following constitutes a practical method of diago-
nalizing an n x n hermitian matrix A by means of a unitary
matrix. For each eigenvalue find a basis for the correspond-
ing eigenspace. Then apply the Gram-Schmidt procedure to
get an orthonormal basis of each eigenspace. These bases are
then combined to form an orthonormal set, say {Xi,..., X n}.
n
By 9.1.4 this will be a basis of C . If U is the matrix with
columns X\,..., X n, then U is hermitian and U*AU is diago-
nal, as was shown in the discussion preceding 9.1.2. The same
procedure is effective for real symmetric matrices.

Example 9.1.1
Find a real orthogonal matrix which diagonalizes the matrix

The eigenvalues of A are 3 and —1, (real of course), and
corresponding eigenvectors are

a n d
( l ) ( ~ l ) '