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

Find an eigenbasis for the given matrice and diagonalize:
_{$A = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$}

The eigenbasis for the given matrice is $[\begin{matrix} 0 & 0 \\ 0 & 2 \end{matrix}]$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Solving the given matrices

We solve:

$d e t (A - λ l) = 0 |\begin{matrix} 1 - λ & 1 \\ 1 & 1 - λ \end{matrix}| = 0 {(1 - λ)}^{2} - 1 = 0 λ^{2} - 2 λ = 0 λ (λ - 2) = 0 λ_{1} = 0, λ_{2} = 2$

Step 2: Solving by different values of λ

For $λ = 0$ , we solve:

$[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] [\begin{matrix} x_{1} \\ y_{1} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] x_{1} + y_{1} = 0$

We can choose an eigenvector:

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

For $λ = 2$ , we solve:

$[\begin{matrix} - 1 & 1 \\ 1 & - 1 \end{matrix}] [\begin{matrix} x_{1} \\ y_{1} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] x_{1} - y_{1} = 0$

We can choose an eigenvector:

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

Now, localid="1659533233518" $\{v_{1}, v_{2}\}$ is an eigenbasis for $R^{2}$ , therefore the diagonalization of A in the eigenbasis is $[\begin{matrix} 0 & 0 \\ 0 & 2 \end{matrix}]$ _.

Hence the final answer is $[\begin{matrix} 0 & 0 \\ 0 & 2 \end{matrix}]$ .

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Find an eigenbasis for the given matrice and diagonalize:
_{$A = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$}

Step by Step Solution

Step 1: Solving the given matrices

Step 2: Solving by different values of λ

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

Find an eigenbasis for the given matrice and diagonalize:A=[1111]

Step by Step Solution

Step 1: Solving the given matrices

Step 2: Solving by different values of λ

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

Find an eigenbasis for the given matrice and diagonalize:
_{$A = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$}