Q39E Consider an n脳nmatrix such that... [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: Q39E (page 338)

Consider an $n 脳 n$ matrix such that the sum of the entries in each row is . Show that the vector
$\overset{鈫�}{e} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}]$
In $R^{n}$ is an eigenvector of A. What is the corresponding eigenvalue?

Short Answer

Expert verified

The eigenvalue of the matrix A with the vector $\overset{鈫�}{e} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}]$ is $位 = 1$

Step by step solution

Finding the ithcoordinate of Ae

For an $n 脳 n$ matrix A and scalar $位, 位$ , is an eigenvalue of A, if there exist a non-zero vector $\overset{鈫�}{v}$ in $R^{n}$ such that,

$A \overset{鈫�}{v} = 位 \overset{鈫�}{v} or A \overset{鈫�}{v} - 位 \overset{鈫�}{v} = 0 or A \overset{鈫�}{v} - 位 \overset{鈫�}{l_{n} v} = \overset{鈫�}{0} or (A - {位濒}_{n}) \overset{鈫�}{v} = \overset{鈫�}{0}$

For the given vector $\overset{鈫�}{e} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}]$ in the matrix A,

The $i^{th}$ coordinate of Ae will be,

$1 脳 a_{i, j} + 1 脳 a_{i, 2} + . . . + 1 脳 a_{i, n} = 1 脳 {鈭�}_{j - 1}^{n} a_{i, j} = 1 脳 1 = 1$

Finding the eigenvalue λ

So the vector will be,

$Ae = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}] = e$

Therefore, e is an eigenvalue of Matrix A, with the corresponding eigenvalue

Thus, the eigenvalue of the matrix A with the vector $\overset{鈫�}{e} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}]$ is $位 = 1$

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91影视

Consider an $n 脳 n$ matrix such that the sum of the entries in each row is . Show that the vector
$\overset{鈫�}{e} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}]$
In $R^{n}$ is an eigenvector of A. What is the corresponding eigenvalue?

Short Answer

Step by step solution

Finding the ithcoordinate of Ae

Finding the eigenvalue λ

One App. One Place for Learning.

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91影视

Consider an n脳nmatrix such that the sum of the entries in each row is . Show that the vectore鈫�=[11..1]In Rnis an eigenvector of A. What is the corresponding eigenvalue?

Short Answer

Step by step solution

Finding the ithcoordinate of Ae

Finding the eigenvalue λ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Decision Maths

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Consider an $n 脳 n$ matrix such that the sum of the entries in each row is . Show that the vector
$\overset{鈫�}{e} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}]$
In $R^{n}$ is an eigenvector of A. What is the corresponding eigenvalue?