Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Short Answer

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.
For each of the matrices A in Exercise 1 through 20 ,find all (real) eigenvalues. Then find a basis of each eigenspaces ,and diagonalize A, if you can. Do not use technology.
$[\begin{matrix} 6 & 3 \\ 2 & 7 \end{matrix}]$

Given eigenvalue, find a basis of the associated eigensspace.

$\det (A - λl) = 0$

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

$[\begin{matrix} 4 & 0 \\ 0 & 9 \end{matrix}]$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: definition of matrices

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

Given,

$d e t (A - λ l) = 0 [\begin{matrix} 6 - λ & 3 \\ 2 & 7 - λ \end{matrix}] = 0 (6 - λ) (7 - λ) - 6 = 0 λ^{2} - 13 λ + 36 = 0 (λ - 4) (λ - 9) = 0 λ_{1} = 4, λ_{2} = 9$

We solved,

$λ = 4 d e t (A - 4 l) x = 0 [\begin{matrix} 2 & 3 \\ 2 & 3 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] 2 x_{1} + 3 x_{2} = 0$

Basic of this eigenspace is

$\{[\begin{matrix} 3 \\ - 2 \end{matrix}]\} = : \{v_{1}\}$

Step 2: multiply the matrices

Similarly $λ = 9 (A - 9 l) x = 0 [\begin{matrix} - 3 & 3 \\ 2 & - 2 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] - 3 x_{1} + 3 x_{2} = 0, 2 x_{1} - x_{2} = 0 x_{1} - x_{2} = 0$

Basic of Eigen space is

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

Hence,

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

$[\begin{matrix} 4 & 0 \\ 0 & 9 \end{matrix}]$

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: definition of matrices

Step 2: multiply the matrices

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: definition of matrices

Step 2: multiply the matrices

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths