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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: Q7-56E (page 384)

TRUE OR FALSE
56. If $\overset{鈫�}{u}$ is a nonzero vector in ${鈩�}^{n}$ , then $\overset{鈫�}{u}$ must be an eigenvector of matrix $\overset{鈫�}{u} {\overset{鈫�}{u}}^{T}$ .

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Define eigenvector

For any scaler $位$ , if $A \overset{鈫�}{v} = 位 \overset{鈫�}{v}$ , then $\overset{鈫�}{v}$ is the eigenvector of matrix A.

02

Determine the given statement is true or false

Let, $U = u u^{T}$ .

We have, $u_{i, j} = u_{i} u_{j}$ .

Let, $v = u u^{T} u$ .

Then,

$v_{i} = {鈭�}_{j = 1}^{n} u_{i, j} u_{j} = {鈭�}_{j = 1}^{n} u_{i} u_{j}^{2} = u_{i} {鈭�}_{j = 1}^{n} u_{j}^{2}$

Which implies, v is a multiple of u.

Therefore, u is an eigenvector of $u u^{T}$ .

Hence, the given statement is true.

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Most popular questions from this chapter

Consider a 4 脳 4 matrix $A = | \begin{matrix} B & C \\ 0 & D \end{matrix} |$ where B, C, and D are 2 脳 2 matrices. What is the relationship among the eigenvalues of A, B, C, and D?

Arguing geometrically, find all eigen vectors and eigen values of the linear transformations. In each case, find an eigen basis if you can, and thus determine whether the given transformation is diagonalizable.

The linear transformation with $T (\overset{鈫�}{v}) = \overset{鈫�}{v}$ , and $T (\overset{鈫�}{w}) = \overset{鈫�}{v} + \overset{鈫�}{w}$ for the vectors $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ in $R^{2}$ sketched below.

Find a basis of the linear space V of all $2 脳 2$ matrices Afor which both $[\begin{matrix} 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 1 \\ 2 \end{matrix}]$ are eigenvectors, and thus determine the dimension of.

If a vector $\overset{鈬赌}{v}$ is an eigenvector of both Aand B, is $\overset{鈬赌}{v}$ necessarily an eigenvector of A+B?

If $\overset{鈬赌}{v}$ is any nonzero vector in $R^{2}$ , what is the dimension of the space Vof all $2 脳 2$ matrices for which $\overset{鈬赌}{v}$ is an eigenvector?

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