Q1E For a given eigenvalue, find a b... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q1E (page 345)

For a given eigenvalue, find a basis of the associated eigenspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.
For each of the matrices A in Exercise1 through 20 , find all (real)eigenvalues. Then find a basis of each eigenspaces, and diagonalize A, if you can. Do not use technology.
$[\begin{matrix} 7 & 8 \\ 0 & 9 \end{matrix}]$

Short Answer

Expert verified

For given eigenvalue, $d e t (A - 位 l) = 0, \{v_{1}, v_{2}\}$ , is an eigenbasis for $R^{2}$ ,so the diagonalization of A in this eigenbasis is $[\begin{matrix} 7 & 0 \\ 0 & 9 \end{matrix}]$ .

Step by step solution

Definition of matrices

A function is defined as a relationship between a set of inputs that each have one output.

Given,

$d e t (A - 位 l) = 0 [\begin{matrix} 7 - 位 & 8 \\ 0 & (9 - 位) \end{matrix}] = 0 (7 - 位) (9 - 位) 位_{1} = 7, 位_{2} = 9$

$位 = 7$ we solve,

$(A - 7 l) x = 0 [\begin{matrix} 0 & 8 \\ 0 & 2 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] 8 x_{2} = 0, 2 x_{2} = 0, x_{2} = 0$

Basic of this eigenspace is

$\{[\begin{matrix} 1 \\ 0 \end{matrix}]\} = : \{v_{1}\}$ .

Multiply the matrices

We solve:

$位 = 9 (A - 9 l) x = 0 [\begin{matrix} - 2 & 8 \\ 0 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] - 2 x_{1} + 8 x_{2} = 0 x_{1} - 4 x_{2} = 0$

Basic of this eigenspace is

$\{[\begin{matrix} 4 \\ 1 \end{matrix}]\} = : \{v_{2}\}$

Hence,

$\{v_{1}, v_{2}\}$ is an eigenbasis for $R^{2}$ , so the diagonalization of A in this eigenbasis is

$[\begin{matrix} 7 & 0 \\ 0 & 9 \end{matrix}]$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Definition of matrices

Multiply the matrices

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Mechanics Maths

Theoretical and Mathematical Physics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.