/*! 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} Q63E Consider a symmetric invertible ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q63E (page 235)

Consider a symmetric invertible n脳nmatrix Awhich admits an LDU-factorization A=LDU. See Exercises 90, 93, and 94 of Section 2.4. Recall that this factorization is unique. See Exercise 2.4.94. Show that

U=LT (This is sometimes called theLDLT - factorizationof a symmetric matrix A.)

Short Answer

Expert verified

Because of LDU factorizationU=LT andL=UT claimed.

Step by step solution

01

Properties of the transpose.

Consider the properties of the transpose below.

  1. (A+B)T=AT+BT For all m脳nmatrices Aand B.
  2. (kA)T=kAT For all m脳nmatrices Aand for all scalars k.
  3. (AB)T=BTAT For all m脳pmatrices Aand for all matrices B.
  4. rank(AT)=rank(A) For all matrices A.
  5. (AT)-1=(A-1)T For all invertible n脳nmatrices A.

By Exercise 2.4.94b, the given LDU factorization of A is unique.

And by the properties of transpose consider the term below.

A=AT=LDU=UTDTLT=UTDLT

This is another way to write the LDU factorization of A.

Since,UTis lower triangular and is an upper triangular.

Hence, by the uniqueness of the LDU factorizationU=LTandL=UTclaimed.

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 a QRfactorization

M=QRShow that R=QTM.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

8.[5422],[367-2]

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in exercises 1 through 14.

1.[21-2]

Consider a basisv鈫�1,v鈫�2,...v鈫�m of a subspaceVofrole="math" localid="1659434380505" 鈩�n. Show that a vectorx鈫� inrole="math" localid="1659434402220" 鈩�nis orthogonal toV if and only ifx鈫� is orthogonal to all vectorsv鈫�1,v鈫�2,...v鈫�m.

Question: Consider an n脳mmatrix A, a vector in Rm, and a vector w鈫�in Rm. Show that (Av鈫�).w鈫�=v鈫�.(ATw鈫�).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Decision 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.