Elementary Linear Algebra, 11th Edition

Chapter 7Diagonalization and Quadratic Forms

CHAPTER CONTENTS

1. 7.1 Orthogonal Matrices
2. 7.2 Orthogonal Diagonalization
3. 7.3 Quadratic Forms
4. 7.4 Optimization Using Quadratic Forms
5. 7.5 Hermitian, Unitary, and Normal Matrices

INTRODUCTION

In Section 5.2 we found conditions that guaranteed the diagonalizability of an $n\times n$

7.1 Orthogonal Matrices

In this section we will discuss the class of matrices whose inverses can be obtained by transposition. Such matrices occur in a variety of applications and arise as well as transition matrices when one orthonormal basis is changed to another.

Orthogonal Matrices

We begin with the following definition.

Recall from Theorem 1.6.3 that if either product in (1) holds, then so does the other. Thus, $A$either. $A{A}^T=I\text{ or }{A}^{T}A=I$

DEFINITION 1 A square matrix $A$orthogonal if its transpose is the same as its inverse, that is, if

$$A^{-1}=A^T$$

or, equivalently, if

(1)$$\begin{array}{c}A{A}^T=A^{T}A=I\end{array}$$

EXAMPLE 1 A $3\times 3$

The matrix

$$A=\left[\begin{array}{lll}\frac{3}{7}\hfill & \frac{2}{7}\hfill & \frac{6}{7}\hfill \\ -\frac{6}{7}\hfill & \frac{3}{7}\hfill & \frac{2}{7}\hfill \\ \frac{2}{7}\hfill & \frac{6}{7}\hfill & -\frac{3}{7}\hfill \end{array}\right]$$

is orthogonal since

$$A^{T}A=\left[\begin{array}{lll}\frac{3}{7}\hfill & -\frac{6}{7}\hfill & \frac{2}{7}\hfill \\ \frac{2}{7}\hfill & \frac{3}{7}\hfill & \frac{6}{7}\hfill \\ \frac{6}{7}\hfill & \frac{2}{7}\hfill & -\frac{3}{7}\hfill \end{array}\right]\left[\begin{array}{lll}\frac{3}{7}\hfill & \frac{2}{7}\hfill & \frac{6}{7}\hfill \\ -\frac{6}{7}\hfill & \frac{3}{7}\hfill & \frac{2}{7}\hfill \\ \frac{2}{7}\hfill & \frac{6}{7}\hfill & -\frac{3}{7}\hfill \end{array}\right]=\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

EXAMPLE 2 Rotation and Reflection Matrices Are Orthogonal

Recall from Table 5 of Section 4.9 that the standard matrix for the counterclockwise rotation of $R^2$$\text{angle }\theta$

$$A=\left[\begin{array}{ll}\text{cos}\theta \hfill & -\text{sin}\theta \hfill \\ \text{sin}\theta \hfill & \text{cos}\theta \hfill \end{array}\right]$$

This matrix is orthogonal for all choices of $\theta$

$$A^{T}A=\left[\begin{array}{ll}\text{cos }\theta \hfill & \text{sin }\theta \hfill \\ -\text{sin }\theta \hfill & \text{cos }\theta \hfill \end{array}\right][\begin{array}{c}\cos \end{array}$$