DEFINITION Let A be an n x n matrix. Any values of A for which Av = Av has nontrivial solutions v are called eigenvalues of A. The corresponding nonzero vectors v are called eigenvectors of A. 1 = 10, v = c1(1, -4, 0) + c2(0, 0, 1), 6 -1 0 -16 -4 -1 A = where ci and c2 are con- 10 stants.

DEFINITION Let A be an n x n matrix. Any values of A for which Av = Av has nontrivial solutions v are called eigenvalues of A. The corresponding nonzero vectors v are called eigenvectors of A. 1 = 10, v = c1(1, -4, 0) + c2(0, 0, 1), 6 -1 0 -16 -4 -1 A = where ci and c2 are con- 10 stants.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 26EQ

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Question

use Equation in the picture to verify that λ and v are an eigenvalue/eigenvector pair for the given matrix A.

DEFINITION
Let A be an n x n matrix. Any values of A for which
Av = Av
has nontrivial solutions v are called eigenvalues of A. The corresponding nonzero
vectors v are called eigenvectors of A.

1 = 10, v = c1(1, -4, 0) + c2(0, 0, 1),
6 -1 0
-16
-4 -1
A =
where ci and c2 are con-
10
stants.

Expert Solution

Step 1

We have given a matrix, $A = [\begin{matrix} 6 & - 1 & 0 \\ - 16 & 6 & 0 \\ - 4 & - 1 & 10 \end{matrix}] .$

Any values $λ$ for which $A v = λ v$ has nontrivial solutions $v$ are called eigenvalues of $A .$ the corresponding nonzero vectors $v$ are called eigenvectors of $A .$

Here $λ = 10, v_{1} = [\begin{matrix} 1 \\ - 4 \\ 0 \end{matrix}], v_{2} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] .$

We need to verify $λ$ and $v_{1}, v_{2}$ are eigenvalue eigenvectors of given matrix.

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