Q49E The eigenvalues of a symmetric m... [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 1: Q49E (page 1)

The eigenvalues of a symmetric matrix $A$ must be equal to the singular values of $A$ .

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

Find the eigenvalues

Consider a matrix

$A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$

Determine the eigenvalues as follows:

$(A - 位濒) = 0$

$[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] - [\begin{matrix} 位 & 0 \\ 0 & 位 \end{matrix}] = [\begin{matrix} 1 - 位 & 1 \\ 0 & - 1 - 位 \end{matrix}] (1 - 位) (- 1 - 位) = 0 - 1 - 位 + 位 + 位^{2} = 0 位^{2} - 1 = 0 位 = 卤 1$

Thus, the eigenvalues are $1, - 1$ .

Find the singular values.

Find $A^{T} of A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ as follows:

$A^{T} [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$

Find $A^{T} A$ as follows:

$A^{T} A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] A^{T} A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Determine the singular values as follows:

$(A^{T} A - 位滨) = 0$

$[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] - [\begin{matrix} 位 & 0 \\ 0 & 位 \end{matrix}] = [\begin{matrix} 1 - 位 & 1 \\ 0 & 1 - 位 \end{matrix}]$

$(1 - 位) (1 - 位) = 0 {(1 - 位)}^{2} = 0 {(- 位 + 1)}^{2} = 0 位 = 1$

The singular values are found from the non-zero eigenvalues of $A^{T} A$ .

The singular value is $蟽 = \sqrt{位}$ .

Thus, the singular value of A is 1.

Final Answer

In view of the above example, the eigenvalues of a symmetric matrix A are not equal to A's singular value.

The given statement is FALSE.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

The eigenvalues of a symmetric matrix $A$ must be equal to the singular values of $A$ .

Short Answer

Step by step solution

Find the eigenvalues

Find the singular values.

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

The eigenvalues of a symmetric matrix Amust be equal to the singular values ofA.

Short Answer

Step by step solution

Find the eigenvalues

Find the singular values.

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

The eigenvalues of a symmetric matrix $A$ must be equal to the singular values of $A$ .