/*! 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} Q 5.5-6E Question: a. Consider an n聽脳聽... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q 5.5-6E (page 260)

Question: a. Consider an n脳mmatrix Pand an m脳nmatrix Q. Show that

trace( PQ )=trace( QP ).

b.Compare the following two inner products in R苍脳尘

鈱�A,B鈱�=trace (ATB)

and

鈱�A,B鈱�=trace (ABT)

See Example 3 and Exercises 4 and 5.

Short Answer

Expert verified
  1. trace ( PQ )=trace ( QP ).
  2. A,B=A,B

Step by step solution

01

Consider for part (a).

Here P and Q are matrices of size n脳 mand m脳 n respectively.

To show that trace ( PQ ) = trace ( QP ) . Observe that below

(PQ)ij=鈭�k=1mpikqkjtrace(PQ)=鈭�i=1n(PQ)ii=鈭�i=1n鈭�k=1mpikqki

Similarly,

trace(QP)=鈭�i=1n(QP)ii=鈭�i=1m鈭�k=1nqikpki=鈭�i=1m鈭�k=1nqkipik

By interchanged the role of I and k.

=trace(QP)

Thus, trace( PQ ) = trace( QP ) .

02

Consider for the part (b).

Now to prove that A,B=A,Bobserve that

A,B=trace(ABT)=trace(BAT)=trace((BAT)T)=trace(ABT)=A,B

Thus, we showed that A,B=A,B

Hence,

  1. trace ( PQ ) = trace ( QP ) .
  2. A,B=A,B

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

Consider an m脳n matrix A withKer(A)={0鈫�}. Show that there exists ann脳m matrix B such thatBA=ln.

Complete the proof of Theorem 5.1.4: Orthogonal projection is linear transformation.

If Ais ann脳mmatrix, is the formulaim(A)=im(AAT)necessarily true? Explain.

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -B.

a.Consider the matrix product Q1=Q2S, where both Q1and Q2 are n脳mmatrices with orthonormal columns. Show that Sis an orthogonal matrix. Hint: Computelocalid="1659499054761" Q1T=Q1=(Q2S)TQ2S. Note that

Q1T=Q=Q2T=Q2=Im

b.Show that the QRfactorization of an n脳mmatrix Mis unique. Hint: IfQ1R1=Q2R2, thenQ1=Q2R2R1-1 . Now use part (a) and Exercise 50a.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.