Page 566 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

7.3 In this section we shall be concerned with the problem of finding an

ORTHOGONAL orthonormal basis for with the Euclidean inner product consisting of
DIAGONALIZATION
eigenvectors of a given matrix A. Our earlier work on symmetric

matrices and orthogonal matrices will play an important role in the

discussion that follows.

Orthogonal Diagonalization Problem

As in the preceding section, we begin by stating two problems. Our goal is to show that the problems are equivalent.

The Orthonormal Eigenvector Problem Given an matrix A, does there exist an orthonormal basis for with the
Euclidean inner product that consists of eigenvectors of the matrix A?

The Orthogonal Diagonalization Problem (Matrix Form) Given an matrix A, does there exist an orthogonal matrix P

such that the matrix is diagonal? If there is such a matrix, then A is said to be orthogonally diagonalizable

and P is said to orthogonally diagonalize A.

For the latter problem, we have two questions to consider:

Which matrices are orthogonally diagonalizable?

How do we find an orthogonal matrix to carry out the diagonalization?

With regard to the first question, we note that there is no hope of orthogonally diagonalizing a matrix A unless A is

symmetric (that is, ). To see why this is so, suppose that

where P is an orthogonal matrix and D is a diagonal matrix. Since P is orthogonal, (1)
be written as , so it follows that 1 can

(2)

Since D is a diagonal matrix, we have . Therefore, transposing both sides of 2 yields

so A must be symmetric.

Conditions for Orthogonal Diagonalizability

The following theorem shows that every symmetric matrix is, in fact, orthogonally diagonalizable. In this theorem, and for
the remainder of this section, orthogonal will mean orthogonal with respect to the Euclidean inner product on .