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Orthogonal Diagonalization

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Introduction

Today we will examine the problem of finding an orthonormal

basis for R n^ consisting of eigenvectors of a given n × n matrix.

It will turn out that suitable n × n matrices can be diagonalized

by orthogonal matrices.

Some Partial Answers

Suppose A is orthogonally diagonalizable, then

D = PT^ AP = DT

which implies that A = PDPT^ and AT^ = PDPT^ , in other words A is symmetric.

The converse of this statement is also true.

Some Partial Answers

Suppose A is orthogonally diagonalizable, then

D = PT^ AP = DT

which implies that A = PDPT^ and AT^ = PDPT^ , in other words A is symmetric.

The converse of this statement is also true.

Equivalent Statements

Theorem

If A is an n × n matrix, then the following are equivalent.

(^1) A is orthogonally diagonalizable.

(^2) A has an orthonormal set of n eigenvectors.

(^3) A is symmetric.

Proof.

( 1 ) =⇒ ( 2 ) =⇒ ( 1 ) =⇒ ( 3 )

Equivalent Statements

Theorem

If A is an n × n matrix, then the following are equivalent.

(^1) A is orthogonally diagonalizable.

(^2) A has an orthonormal set of n eigenvectors.

(^3) A is symmetric.

Proof.

( 1 ) =⇒ ( 2 ) =⇒ ( 1 ) =⇒ ( 3 )

Equivalent Statements

Theorem

If A is an n × n matrix, then the following are equivalent.

(^1) A is orthogonally diagonalizable.

(^2) A has an orthonormal set of n eigenvectors.

(^3) A is symmetric.

Proof.

( 1 ) =⇒ ( 2 ) =⇒ ( 1 ) =⇒ ( 3 )

Properties of Symmetric Matrices

Theorem

If A is a symmetric matrix, then

(^1) the eigenvalues of A are all real numbers, and

(^2) eigenvectors from different eigenspaces are orthogonal.

Proof.

Diagonalization of Symmetric Matrices

Steps:

(^1) Find a basis for each eigenspace of A.

(^2) Apply the Gram-Schmidt process to each of these bases to

obtain an orthonormal basis for each eigenspace.

(^3) Form the matrix P from the basis vectors found in step (2).

Example

Example

Find an orthogonal matrix P which diagonalizes

A =

[

]

Eigensystems:

Eigenvalue Eigenvector

2 ( 1 , 2 )

7 (− 2 , 1 )

Example

Example

Find an orthogonal matrix P which diagonalizes

A =

Eigensystems:

Eigenvalue Eigenvector

3 (− 1 , 0 , 1 )

3 (− 1 , 1 , 0 ) 0 ( 1 , 1 , 1 )

Example

Example

Find an orthogonal matrix P which diagonalizes

A =

Eigensystems:

Eigenvalue Eigenvector

3 (− 1 , 0 , 1 )

3 (− 1 , 1 , 0 ) 0 ( 1 , 1 , 1 )

Example

Example

Find an orthogonal matrix P which diagonalizes

A =

Eigensystems:

Eigenvalue Eigenvector

4 +

Homework

Read Section 7.3 and work exercises 1abe, 3, 5, 10, 11.