Q36E Consider an invertible n 脳 n ma... [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 8: Q36E (page 401)

Consider an invertible n 脳 n matrix A. What is the relationship between the matrix R in the QR factorization of A and the matrix L in the Cholesky factorization of $A^{T} A$ ?

Short Answer

Expert verified

The transpose of the matrix R in the QR - decomposition of A is the matrix L in the Cholesky factorization of $A^{T} A$ .

Step by step solution

Given Information:

Invertible n 脳 n matrix A

Finding the Relationship:

We know that an invertible matrix A of rank n can be uniquely decomposed as A = QR , where Q is an orthogonal matrix (that is, ${QQ}^{T} = l = Q^{T} Q$ ) and R is a positive diagonal upper triangular matrix. Take, for example, $A^{T} A$ .

$A^{T} A = {(QR)}^{T} (QR) = R^{T} Q^{T} QR = R^{T} lR = {LL}^{T}$

A lower triangular matrix with positive elements is $L = R^{T}$ .

In the Cholesky factorization of, $A^{T} A$ the matrix L is the transpose of the matrix R in the QR - decomposition of A.

Determining the Result:

The transpose of the matrix R in the QR - decomposition of A is the matrix L in the Cholesky factorization of $A^{T} A$ .

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91影视

Consider an invertible n 脳 n matrix A. What is the relationship between the matrix R in the QR factorization of A and the matrix L in the Cholesky factorization of $A^{T} A$ ?

Short Answer

Step by step solution

Given Information:

Finding the Relationship:

Determining the Result:

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91影视

Consider an invertible n 脳 n matrix A. What is the relationship between the matrix R in the QR factorization of A and the matrix L in the Cholesky factorization ofATA?

Short Answer

Step by step solution

Given Information:

Finding the Relationship:

Determining the Result:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Consider an invertible n 脳 n matrix A. What is the relationship between the matrix R in the QR factorization of A and the matrix L in the Cholesky factorization of $A^{T} A$ ?