Exercise 22.11. This exercise leads through a direct argument, without reference to Gram-Schmidt, in the case...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Exercise 22.11. This exercise leads through a direct argument, without reference to Gram-Schmidt, in the case of invertible square matrices that QR-decompositions are uniquely determined. That is, if an invertible n x n matrix A is equal to QR and to Q'R' for orthogonal Q and Q' and upper triangular R and R' with positive diagonal entries then Q = Q' and R = R'. -1 (a) Show Q-¹Q' = RR'¯¹, and that the left side is orthogonal and the right side is upper triangular with QR, multiply both sides by Q¹ on positive entries. (Hint for the equality: beginning with Q'R' the left and R¹ on the right.) = (b) Show that if Q−¹Q' = In and RR'¯¹ In then Q Q' and R = R'. (c) Show that if an orthogonal n × n matrix M is also upper triangular with positive diagonal entries then it equals In, and combine this with (a) and (b) to conclude that Q = Q' and R = R'. (Hint: first show the left column of M must be e₁ [and not -e₁, for instance] and then use that knowledge to deduce that the second column must be e2, and so on. Think about n = 2 before thinking about general n.) = - Exercise 22.11. This exercise leads through a direct argument, without reference to Gram-Schmidt, in the case of invertible square matrices that QR-decompositions are uniquely determined. That is, if an invertible n x n matrix A is equal to QR and to Q'R' for orthogonal Q and Q' and upper triangular R and R' with positive diagonal entries then Q = Q' and R = R'. -1 (a) Show Q-¹Q' = RR'¯¹, and that the left side is orthogonal and the right side is upper triangular with QR, multiply both sides by Q¹ on positive entries. (Hint for the equality: beginning with Q'R' the left and R¹ on the right.) = (b) Show that if Q−¹Q' = In and RR'¯¹ In then Q Q' and R = R'. (c) Show that if an orthogonal n × n matrix M is also upper triangular with positive diagonal entries then it equals In, and combine this with (a) and (b) to conclude that Q = Q' and R = R'. (Hint: first show the left column of M must be e₁ [and not -e₁, for instance] and then use that knowledge to deduce that the second column must be e2, and so on. Think about n = 2 before thinking about general n.) = -
Expert Answer:
Answer rating: 100% (QA)
A We are given that A QR and A QR where Q and Q are orthogonal matrices ie QQ I and R and R are uppe... View the full answer
Related Book For
Cost management a strategic approach
ISBN: 978-0073526942
5th edition
Authors: Edward J. Blocher, David E. Stout, Gary Cokins
Posted Date:
Students also viewed these mathematics questions
-
For this network design assignment, you are free to come up with a fictional midsized company engaged in a dot-com business for which you will select the business focus. The company can be a...
-
Phaneendra and Kavitha were trying to come up with some new ideas for their housekeeping service. Phaneendra said, We just dont have enough customers; we need to drop our prices and to do that we...
-
At a recent board meeting, the president and CEO got into a heated argument about whether to shut down the firms plant in Miami. The Miami plant currently loses $60,000 monthly. The president of the...
-
In terms of Federal tax consequences, why are lifetime transfers to charity preferable to testamentary transfers?
-
Examine the set of relations in Figure 4-39. What normal form are these in? How do you know this? If they are in 3NF, convert the relations into an EER diagram. What assumptions did you have to make...
-
Ventaz Corp manufactures small windows for back yard sheds.Historically, its demand has ranged from 30 to 50 windows per day with an average of 41. Alex is the one production worker and he works...
-
List the major parts of a complex corporate income statement for WRS Athletic Clubs, Inc., for the year ended December 31, 2007. Include all the major parts of the income statement, starting with net...
-
Refer to the data in Exercise 7-7. Required: 1. Allocate the costs of the support departments using the sequential method. 2. Using direct labor hours, compute departmental overhead rates for the two...
-
PT ABC purchased new equipment that will be used to make computer processors. in the first year of operation of its new equipment, PT ABC will sell 1 billion worth of chips, with a gross profit...
-
Felde Bucket Co., a manufacturer of rain barrels, had the following data for 2016: Sales .......................... 2,500 units Sales price ................... $40 per unit Variable costs...
-
Staley Inc. reported the following data: Net income $396,200 Depreciation expense 61,250 Loss on disposal of equipment 27,600 Increase in accounts receivable 9,000 Increase in accounts payable 3,350...
-
1. The political and legal background of European integration in the aftermath of World War II. 2. The European Communities (the European Coal and Steel Community, the European Economic Community,...
-
A motorist drives north for 38.0 minutes at 95.5 km/h and then stops for 15.0 minutes. He then continues north, traveling 130 km in 2.00 h. (a) What is his total displacement? 190 km (b) What is his...
-
A dealer wants to create a list of classic cars in her showroom. Using the existing Car class in this exercise. Create a static ArrayList called showroom that adds a Car to the showroom whenever a...
-
Royal Lawncare Company produces and sells two packaged products-Weedban and Greengrow. Revenue and cost information relating to the products follow: Selling price per unit Variable expenses per unit...
-
A vertical tube open at the top contains 2.4 cm of oil (density 0.82 g/cm) floating on 2.8 cm of water. Part A Find the gauge pressure at the bottom of the tube. Express your answer in pascals and...
-
Determine whether the desktops are on hardware or cloud-based server. Also, specify the version of the OS server. 127 desktops with Windows 10 Enterprise 14 desktops with Mac 11 Big Sur 25 desktops...
-
What are the two components of a company's income tax provision? What does each component represent about a company's income tax provision?
-
Eastern Company manufactures special electrical equipment and parts. The company uses a standard cost system with separate standards established for each product. The transformer department...
-
Phelps Inc. manufactures several different types of candy for various retail stores. The accounting manager has requested that you determine the sales dollars required to break even for next quarter...
-
Obtain from your library a copy of following article: Robert S. Kaplan and David P. Norton, Mastering the Management System, Harvard Business Review (January 2008), pp. 6377. The authors of this...
-
Explain the limitations that auditors face when they perform the attest function.
-
Indicate whether you think the following third-party groups would normally represent a (1) primary beneficiary, (2) foreseen party, or (3) foreseeable party. Give reasons for your answers. a. A...
-
Use and Dispose Company was organized to manufacture and sell inexpensive golf clubs that can be used during the golf round and then thrown away at the end of the round. In order to keep the business...
Study smarter with the SolutionInn App