Q40E Consider an invertible n x n mat... [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 5: Q40E (page 225)

Consider an invertible n x n matrix A whose columns are orthogonal, but not necessarily orthonormal. What does the QR factorization of A look like?

Short Answer

Expert verified

The matrix $Q = [\frac{{\overset{鈬赌}{v}}_{1}}{||{\overset{鈬赌}{v}}_{1}||} . . . . \frac{{\overset{鈬赌}{v}}_{n}}{||{\overset{鈬赌}{v}}_{n}||}]$ and $R = [\begin{matrix} ||{\overset{鈬赌}{v}}_{1}|| & 鈥� & 0 \\ 鈰� & 鈰� & 鈰� \\ 0 & 鈥� & ||{\overset{鈬赌}{v}}_{n}|| \end{matrix}]$ .

Step by step solution

QR factorization of matrix

In QR factorization of a matrix A, the matrix is decomposed into a product A = QR, where is an orthonormal matrix and R is an upper triangular matrix.

Describe how QR factorization looks like

Let the matrix A has the columns ${\overset{鈬赌}{v}}_{1}, {\overset{鈬赌}{v}}_{2}, {\overset{鈬赌}{v}}_{3}, . . ., {\overset{鈬赌}{v}}_{n},$ , where the columns are linearly independent.

Then an orthogonal matrix Q, in the factorization of QR is given below:

role="math" localid="1660308888953" $Q = [\frac{{\overset{鈬赌}{v}}_{1}}{||{\overset{鈬赌}{v}}_{1}||} . . . . \frac{{\overset{鈬赌}{v}}_{n}}{||{\overset{鈬赌}{v}}_{n}||}]$

The upper triangular matrix R in the QR factorization is given by:

role="math" localid="1660308922734" $R = [\begin{matrix} ||{\overset{鈬赌}{v}}_{1}|| & 鈥� & 0 \\ 鈰� & 鈰� & 鈰� \\ 0 & 鈥� & ||{\overset{鈬赌}{v}}_{n}|| \end{matrix}]$

Thus, the QR factorization of columns A looks like role="math" localid="1660308963377" $Q R = [\frac{{\overset{鈬赌}{v}}_{1}}{||{\overset{鈬赌}{v}}_{1}||} . . . . \frac{{\overset{鈬赌}{v}}_{n}}{||{\overset{鈬赌}{v}}_{n}||}] [\begin{matrix} ||{\overset{鈬赌}{v}}_{1}|| & 鈥� & 0 \\ 鈰� & 鈰� & 鈰� \\ 0 & 鈥� & ||{\overset{鈬赌}{v}}_{n}|| \end{matrix}]$ .

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Consider an invertible n x n matrix A whose columns are orthogonal, but not necessarily orthonormal. What does the QR factorization of A look like?

Short Answer

Step by step solution

QR factorization of matrix

Describe how QR factorization looks like

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Geometry

Mechanics Maths

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.