Q29E Consider an聽n脳n matrix such th... [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: Q29E (page 337)

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

Step by step solution

Finding the ith coordinate 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,

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

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,

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

Finding the eigenvalue λ

So the vector will be,

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

Therefore, e is an eigenvalue of Matrix A, with the corresponding eigenvalue $位 = 1$

Thus, the eigenvalue of the matrix A with the vectorrole="math" localid="1659588113834" $\overset{鈫�}{e} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}]$ 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 ith coordinate of Ae

Finding the eigenvalue λ

One App. One Place for Learning.

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91影视

Consider ann脳n matrix such that the sum of the entries in each row is . Show that the vectore鈫�=11..1InRn is an eigenvector of A. What is the corresponding eigenvalue?

Short Answer

Step by step solution

Finding the ith coordinate 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

Pure Maths

Calculus

Probability and Statistics

Statistics

Decision Maths

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?