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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 8: Q25E (page 400)

Consider a quadratic form
$q (\overset{鈬赌}{x}) = \overset{鈬赌}{x} . A \overset{鈬赌}{x}$
where A is a symmetricnxnmatrix. Let $\overset{鈬赌}{v}$ be a unit eigenvector of A, with associated eigenvalue $位$ . Find $q (\overset{鈬赌}{v})$ .

Short Answer

Expert verified

$q (\overset{鈬赌}{v}) = 位$

Step by step solution

01

Given Information:

$q (\overset{鈬赌}{x}) = \overset{鈬赌}{x} . A \overset{鈬赌}{x}$

02

Determining q(v⇀):

Take a look at the quadratic form:

$q (\overset{鈬赌}{x}) = \overset{鈬赌}{x} . A \overset{鈬赌}{x}$

A is a nxn symmetric matrix. Let $\overset{鈬赌}{v}$ be the unit eigenvector of A, with lambda as the eigenvalue. Now, use the following formula to access $q (\overset{鈬赌}{v})$ :

role="math" localid="1659615170927" $q (\overset{鈬赌}{v}) = \overset{鈬赌}{v} . A \overset{鈬赌}{v} = \overset{鈬赌}{v} . 位 \overset{鈬赌}{v} = 位 (\overset{鈬赌}{v} . \overset{鈬赌}{v}) = 位 (1)$

(According to the eigen value definition)

$鈬� q (\overset{鈬赌}{v}) = 位$

03

Determining the Result:

$q (\overset{鈬赌}{v}) = 位$

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Most popular questions from this chapter

For each of the quadratic forms q listed in Exercises 1 through 3, find the matrix of q.

3. $(x_{1}, x_{2}, x_{3}) = 3 脳_{1}^{2} + 4 脳_{2}^{2} + 5 脳_{3}^{2} + 6 x_{1} x_{3} + 7 x_{2} x_{3}$

If Ais an invertible symmetric matrix, then $A^{2}$ must be positive definite.

43. If Ais indefinite, then 0 must be an eigenvalue of A.

Consider a singular value decomposition $A = U 危 V^{T}$ of an $n 脳 m$ matrix Awith $rank A = m$ . Let ${\overset{鈫�}{v}}_{1}, 鈥�, {\overset{鈫�}{v}}_{m}$ be the columns of Vand ${\overset{鈫�}{u}}_{1}, 鈥�, {\overset{鈫�}{u}}_{n}$ the columns of U. Without using the results of Chapter 5 , compute ${(A^{T} A)}^{- 1} A^{T} {\overset{鈫�}{u}}_{i}$ . Explain the result in terms of leastsquares approximations.

If the singular values of an $n 脳 n$ matrix A are all 1 , A is necessarily orthogonal?

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