Q50E Find an eigenbasis of given matr... [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: Q50E (page 325)

Find an eigenbasis of given matrix and diagonalize it.
$A = (\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix})$

Short Answer

Expert verified

Eigen basis are $v_{1} = [\begin{matrix} 3 \\ - 1 \end{matrix}]$ and $v_{2} = [\begin{matrix} 1 \\ 2 \end{matrix}]$

The diagonalization of this matrix is $(\begin{matrix} 0 & 7 \\ 0 & 0 \end{matrix})$ .

Step by step solution

Definition of diagonalizable

The matrix A is diagonalizable if there exists an eigenbasis for A. The ${\overset{鈫�}{v}}_{1}, . . . . ., {\overset{鈫�}{v}}_{n}$ is an eigenbasis for A, with $A {\overset{鈫�}{v}}_{1} = 位_{1} {\overset{鈫�}{v}}_{1}, . . ., A {\overset{鈫�}{v}}_{n} = 位_{n} {\overset{鈫�}{v}}_{n}$ , then the matrices

role="math" localid="1659532048605" $S = [\begin{matrix} | & | & | & | \\ {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{v}}_{2} & . . . . & {\overset{鈫�}{v}}_{n} \\ | & | & | & | \end{matrix}]$ androle="math" localid="1659532177343" $B = [\begin{matrix} 位_{1} & 0 & 鈥� & 0 \\ 0 & 位_{2} & 鈥� & 0 \\ 鈰� & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈰� & 位_{n} \end{matrix}]$

Will be diagonalize A, meaning that $S^{- 1} AS = B$

Finding the eigenvalues of the given matrix

$d e t (A - 位 l) = 0 (\begin{matrix} 1 - 位 & 3 \\ 2 & 6 - 位 \end{matrix}) = 0 位^{2} - 7 位 = 0 位_{1} = 0, 位_{2} = 7$

Finding eigenvectors

For $位 = 0$ ,

So the eigenvector,

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

For $位 = 7$ ,

$[\begin{matrix} - 6 & 3 \\ 2 & - 1 \end{matrix}] [\begin{matrix} x_{2} \\ y_{2} \end{matrix}] = 0 2 x_{2} - y_{2} = 0$

We can choose the eigenvector $v_{2} = [\begin{matrix} 1 \\ 2 \end{matrix}]$ .

Now $(v_{1}, v_{2})$ is an eigen basis.

Finding the diagonizable form of the matrix

Diagonalization of this matrix is $(\begin{matrix} 0 & 7 \\ 0 & 0 \end{matrix})$

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91影视

Find an eigenbasis of given matrix and diagonalize it.
$A = (\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix})$

Short Answer

Step by step solution

Definition of diagonalizable

Finding the eigenvalues of the given matrix

Finding eigenvectors

Finding the diagonizable form of the matrix

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Find an eigenbasis of given matrix and diagonalize it.A=(1326)

Short Answer

Step by step solution

Definition of diagonalizable

Finding the eigenvalues of the given matrix

Finding eigenvectors

Finding the diagonizable form of the matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

Calculus

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Find an eigenbasis of given matrix and diagonalize it.
$A = (\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix})$