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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q8.1-2E (page 391)

For each of the matrices in Exercises 1 through 6, find an orthonormal eigenbasis. Do not use technology.
2. $[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$

Short Answer

Expert verified

The orthonormal eigenbasis for the given matrix is

$u_{1} = \frac{1}{\sqrt{2}} [\begin{matrix} - 1 \\ 1 \end{matrix}] u_{2} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ 1 \end{matrix}]$

Step by step solution

01

Definition of Eigenvalues

Consider an n x n matrix A and a scalar $位$ . Then $位$ is an eigenvalue of A if (and only if)

$\det (A - 位 I_{n}) = 0$

This is called the characteristic equation (or the secular equation) of matrix A.

02

Find the orthonormal eigenbasis

Find eigenbases using $\det (A - 位 I_{n}) = 0$

$位 = 0, 2$

For $位 = 0$ ,

$[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = x_{2} [\begin{matrix} - 1 \\ 1 \end{matrix}]$

Therefore,

$E_{0} = s p a n [\begin{matrix} - 1 \\ 1 \end{matrix}]$

For $位 = 2$ ,

$[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = x_{2} [\begin{matrix} 1 \\ 1 \end{matrix}]$

Therefore,

$E_{2} = s p a n [\begin{matrix} 1 \\ 1 \end{matrix}]$

Hence, the orthonormal eigenbasis for the given matrix is,

$u_{1} = \frac{1}{\sqrt{2}} [\begin{matrix} - 1 \\ 1 \end{matrix}] u_{2} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ 1 \end{matrix}]$

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Most popular questions from this chapter

If Ais an invertible symmetric matrix, what is the relationship between the definiteness of A and $A^{- 1}$ ?

Find a decomposition $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 蟽_{2} {\overset{鈫�}{u}}_{2} {\overset{鈫�}{v}}_{2}^{T}$

$f o r A = [\begin{matrix} 6 & 2 \\ - 7 & 6 \end{matrix}]$ See Exercise 29 and Example 2.

Find the dimension of the space $Q_{n}$ of all quadratic forms in n variables.

In Exercises 57 through 61, consider a quadratic form on with symmetric matrix A, with the given properties. In each case, describe the level surface $q (\overset{鈫�}{x}) = 1$ geometrically.

61.qis indefinite and det A < 0

For $A = [\begin{matrix} 2 & 11 & 11 \\ 11 & 2 & 11 \\ 11 & 11 & 2 \end{matrix}]$ , find a nonzero vector $\overset{鈫�}{v}$ in ${鈩�}^{3}$ such that $A \overset{鈫�}{v}$ is orthogonal to $\overset{鈫�}{v}$ .

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