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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 7: Q59E (page 325)

find an eigenbasis for the given matrice and diagonalize:
$A = \frac{1}{9} [\begin{matrix} 8 & 2 & - 2 \\ 2 & 5 & 5 \\ - 2 & 4 & 4 \end{matrix}]$
Representing the orthogonal projection onto the plane $x - 2 y + 2 z = 0$

Short Answer

Expert verified

The eigenbasis for the given matrice is_{$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]$ .}

Step by step solution

01

Solving the given matrices:

Every vector $v 鈭� V$ , Where V is the given plane projects onto itself, so we can choose two non-colinear eigenvector:

$v_{1} = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}]$

_And

$v_{2} = [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}]$

With the eigenvalue $位 = 1$

Any vector perpendicular to V will project onto 0, so we can choose an eigrnvector

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

With the eigenvalue $位 = 0$

02

Solving further:

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

$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]$

Hence the final answer is $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]$

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Most popular questions from this chapter

7:For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

$l_{3}$

Find all $2 脳 2$ matrices for which ${\overset{鈬赌}{e}}_{1}$ is an eigenvector.

Find all the polynomials of degree $鈮� 2$ [a polynomial of the form $f (t) = a + b t + c t^{2}$ ] whose graph goes through the points (1,3) and (2,6) , such thatrole="math" localid="1659541039431" $f' (1) = 1$ [where $f' (1)$ denotes the derivative].

Show that 4 is an eigenvalue of $A = [\begin{matrix} - 6 & 6 \\ - 15 & 13 \end{matrix}]$ ,and find all corresponding eigenvectors.

find an eigenbasis for the given matrice and diagonalize:

$A = \frac{1}{9} [\begin{matrix} 7 & 4 & - 4 \\ 4 & 1 & 8 \\ - 4 & 8 & 1 \end{matrix}]$

Representing the reflection about the plane $x - 2 y + 2 z = 0$ .

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