Q53E find an eigenbasis for the given... [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: Q53E (page 325)

find an eigenbasis for the given matrice and diagonalize:
$A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{matrix}]$

Short Answer

Expert verified

The eigenbasis for the given matrice is $[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 14 \end{matrix}]$ _.

Step by step solution

Solving the given matrices

We solve:

$\det (A - 位濒) = 0 |\begin{matrix} 1 - 位 & 2 & 3 \\ 2 & 4 - 位 & 6 \\ 3 & 6 & 9 - 位 \end{matrix}| = 0 (1 - 位) (4 - 位) (9 - 位) + 36 + 36 - 36 (1 - 位) - 9 (4 - 位) - 4 (9 - 位) = 0 - 位^{3} + 14 位^{2} = 0 位^{2} (- 位 + 14 位) = 0 位_{1, 2} = 0, 位_{3} = 14$

Solving by different values of λ

For $位 = 0$ , we solve:

$[\begin{matrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{matrix}] [\begin{matrix} x_{1} \\ y_{1} \\ z_{1} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}] x_{1} + 2 y_{1} + 3 z_{1} = 0$

We can see all the eigenvector $位 = 0$ for are spanned by vectors:

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

For $位 = 14$ , we solve:

$[\begin{matrix} - 13 & 2 & 3 \\ 2 & - 10 & 6 \\ 3 & 6 & - 5 \end{matrix}] [\begin{matrix} x_{3} \\ y_{3} \\ z_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}] - 13 x_{3} + 2 y_{3} + 3 z_{3} = 0, 2 x_{3} - 10 y_{3} + 6 z_{3} + 0, 3 x_{3} + 6 y_{3} - 5 z_{3} = 0$

We can choose an eigenvector:

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

Now, $\{v_{1}, v_{2}, v_{3}\}$ is an eigenbasis for $R^{3}$ , therefore the diagonalization of A in the eigenbasis is

$[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 14 \end{matrix}]$

Hence the final answer is $[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 14 \end{matrix}]$ .

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91影视

find an eigenbasis for the given matrice and diagonalize:
$A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{matrix}]$

Short Answer

Step by step solution

Solving the given matrices

Solving by different values of λ

One App. One Place for Learning.

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91影视

find an eigenbasis for the given matrice and diagonalize:A=[123246369]

Short Answer

Step by step solution

Solving the given matrices

Solving by different values of λ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Pure Maths

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Geometry

Study anywhere. Anytime. Across all devices.

find an eigenbasis for the given matrice and diagonalize:
$A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{matrix}]$