/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q31E If A聽and B聽are 2脳2聽matrices,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Q31E (page 414)

If Aand Bare 2脳2matrices, then the singular values of matrices ABandBAmust be the same.

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

01

Step 1: Definition of a singular value

The diagonal entries of the matrix are singular values, which are arranged in ascending order.

02

Step 2: To Find TRUE or FALSE

Let

A=0100andB=1000

Find the value of AB.

AB=01001000AB=0000

Find the value of BA.

BA=10000100BA=0100

Both the singular values of ABare 0.

BATBA=00100100=0001

The eigenvalues of BATBAare 1 and 0. Therefore, the singular values of BAare 1 and 0.

Thus, 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

Most popular questions from this chapter

Let 位 be a real eigenvalue of an n x n matrix A. Show that

蟽n猢�|位|猢�蟽1,

where蟽1and蟽nare the largest and the smallest singular values of A, respectively.

A Cholesky factorization of a symmetric matrix A is a factorization of the formA=LLTwhere L is lower triangular with positive diagonal entries.

Show that for a symmetricn脳nmatrix A, the following are equivalent:

(i) A is positive definite.

(ii) All principal submatricesrole="math" localid="1659673584599" A(m)of A are positive definite. See

Theorem 8.2.5.

(iii)det(Am)>0form=1,....,n

(iv) A has a Cholesky factorization A=LLT

Hint: Show that (i) implies (ii), (ii) implies (iii), (iii) implies (iv), and (iv) implies (i). The hardest step is the implication from (iii) to (iv): Arguing by induction on n, you may assume that A(n-1)) has a Cholesky factorization A(n-1)=BBT. Now show that there exist a vector x鈫�inRn-1and a scalar t such that

A=[An-1v鈫�v鈫�Tk]=[B0x鈫�T1][BTx鈫�0t]

Explain why the scalar t is positive. Therefore, we have the Cholesky factorization

A=[B0x鈫�Tt][BTx鈫�0t]

This reasoning also shows that the Cholesky factorization of A is unique. Alternatively, you can use the LDLT factorization of A to show that (iii) implies (iv).See Exercise 5.3.63.

To show that (i) implies (ii), consider a nonzero vector, and define

role="math" localid="1659674275565" y鈫�=[x鈫�0M0]

In Rn(fill in n 鈭� m zeros). Then

role="math" localid="1659674437541" x鈫�A(m)x鈫�=y鈫�TAy鈫�>0

True or false? If Ais a symmetric matrix, thenrank(A)=rank(A2)

If Ais an indefiniten脳n matrix, andR is a realn脳mrankn what can you say about the definiteness of the matrixRTAR?

Consider an SVD

A=U鈭�VT

of ann脳mmatrixA . Show that the columns of U form an orthonormal eigenbasis forAAT . What are the associated eigenvalues? What does your answer tell you about the relationship between the eigenvalues ofATAandAAT ? Compare this with Exercise 7.4.57.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.