/*! 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}

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Q94E (page 103)

a. Show that if an invertible n脳nmatrix A admits an LU-factorization, then it admits an LDU factorization. See Exercise 90 d.

b. Show that if an invertible matrix A admits an L DU-factorization, then this factorization is unique. Hint: Suppose that A=L1D1U1=L2D2U2, then U2U-1=D2-1L2-1L1D1is diagonal (why?). Conclude that U2=U1.

Short Answer

Expert verified
  1. It is proved that an invertible matrix A an LDU factorization.
  2. It is proved that the factorization of A is unique.

Step by step solution

01

Provingan invertible n×nn matrix A admits an L DU-factorization (a)

Suppose that A has LU factorization, where A is an invertible matrix.

Then, , where L is the lower triangular matrix and U is the upper triangular matrix.

Now, in the upper triangular matrix by factorizing the diagonal entries from U, we can have the diagonal matrix and an upper triangular matrix U such that its diagonal entries are 1 Hence, A can have LDU factorization.

Hence,A=LDU

02

Provingthe LDU factorization is unique (b)

Now suppose that A has two factorization, where A is an invertible matrix

Let, A=L1D1U1andA=L2D2U2.

Then,

L1D1U1=L2D2U2

Then,

U2U1-1=D2-1L-12L1D1

Now, the matrix D2D1is a diagonal matrix with all the diagonal entries ispositive Therefore U2=U1,. Similarly, we can show that L1=L2and D1=D2.

Hence, the LDU factorization is unique.

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

In this exercise we will verify part (b) of Theorem 2.3.11 in the special case when A is the transition matrix [0.40.30.60.7]andx炉is the distribution vector[10]. [We will not be using parts (a) and (c) of Theorem 2.3.11]. The general proof of Theorem 2.3.11 runs along similar lines, as we will see in Chapter 7.

  1. ComputeA[12]andA[1-1]. WriteA[1-1]as a scalar multiple of the vector[1-1].
  2. Write the distribution vectorx鈫�=[10]as a linear combination of the vectors[12]and[1-1]
  3. Use your answers in part (a) and (b) to writeAx鈫�as a linear combination of the vectors[12]and[1-1]. More generally, write Amx鈫�as a linear combination of vectors[12]and[1-1], for any positive integer m. See Exercise 81.
  4. In your equation in part (c), let got to infinity to find limm鈫�鈭�Amx鈫�. Verify that your answer is the equilibrium distribution for A.

TRUE OR FALSE?

There exists an invertiblen脳n matrix with two identical rows.

If matrix A is invertible, then matrix 5A must be invertible as well.

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises 16through 23 . Show the effect of these transformations on the letter L considered in Example 5 . In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise 13.

23.role="math" localid="1659695358882" [02-20]

Use the concept of a linear transformation in terms of the formula y=Ax鈬赌, and interpret simple linear transformations geometrically. Find the inverse of a linear transformation from localid="1659964769815" 鈩�2to鈩�2to (if it exists). Find the matrix of a linear transformation column by column.

Consider the transformations from鈩�3to鈩�3defined in Exercises 1 through 3. Which of these transformations are linear?

  1. y1=2x2y2=x2+2y3=2x2
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.