The Polar Decomposition: Prove that every invertible matrix A has a polar decomposition, written A = QB,

Question:

The Polar Decomposition: Prove that every invertible matrix A has a polar decomposition, written A = QB, into the product of an orthogonal matrix Q and a positive definite matrix B > 0. Show that if det A > 0, then Q is a proper orthogonal matrix.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Applied Linear Algebra

ISBN: 978-0131473829

1st edition

Authors: Peter J. Olver, Cheri Shakiban

See More Books

Question Posted: Jun 17, 2016 09:08 AM

See More Questions

Show that every square matrix A can be factored as A = RQ, where R is symmetric, positive semidefinite and Q is orthogonal. Every complex number can be written in polar form as z = rei, where r = |z|...
(a) Show that every invertible linear function L: Rn Rn can be represented by the identity matrix by choosing appropriate (and not necessarily the same) bases on the domain and target spaces. (b)...
1. Determinant of an Inverse Prove for invertible matrix A that 2. Do Determinants Commute? Let A and B be any two n ( n matrices. Explain why | AB | = | BA |. 1
Perhaps more surprising to Mr. Pitkin was a proposal by the VP of Marketing to make a major investment in market share by increasing promotional expenditures by $2.5 million during 1998-2000. Sales...
What distinguishes cyber crime from traditional crime?
*28.6 T Ltd has owned 90% of the ordinary share capital of B Ltd for many years. Both are UK resident. Recent results for the two companies are as follows: T Ltd T Ltd B Ltd y/e 31/3/09 y/e 31/3/10...
Refer to Exercises 9.21 and 9.29 (p. 487,493). The SAS simple linear regression printout relating number of employees y to age of a fast-growing firm x is reproduced above. a. Test to determine...
The management of Rousseau Corp. is considering the effects of various inventory-costing methods on its financial statements and its income tax expense. Assuming that the price the company pays for...
Answer all with solutions At September 30, the end of Beijing Company's third quarter the following stockholders' equity accounts are reported. $420,00 Bee. 320, Con tact 13 P value Paid in capital...
Mr B aged 52 years, has earned rupees 75,00,000 out of his business. His ex-wife gifted him a car worth rupees 8 lakh. He spent a total of rupees 20 lakh during a family trip. He won a lottery of 16...
Find all positive definite orthogonal matrices.
Prove that a symmetric matrix is negative definite if and only if all its eigenvalues are negative.
LHpitals Rule is used incorrectly. Describe the error. 3x + 4x + 1 x=2xx2 lim tim x-2 6x +4 2x - T lim x-2-2 3
What's your favourite way to promote and build interpersonalwellness? How do you nurture your relationships? How do you build friendships? How do you show your loved ones that you care? Share your...
What is the best way to summary this formation? IMC Communication Objectives Increase from 0% to 75% of Millennials in the Dallas-Fort Worth area, ages 22 - 34, with a four-year college degree,...
Branding is very important to marketing, that is while most businesses focus on brands that attract customer's interest. Store brand is unique to a particular store compared to national brand . A...
Do you think that your score accurately reflects your global mindset? Why or why not? What, if anything, is missing from the assessment? How do you think that having a higher global mindset will help...
Healthcare is an ever-changing industry that requires healthcare organizations to align with those changes or risk being left behind. With the advances being made in technology, every corner seems to...
For each of the following state whether the market is most accurately described as perfectly competitive, monopoly, oligopoly, or monopolistic competition, and briefly explain your answer. a. Seattle...
What is an access control list?
Suppose that a random sample of 10,000 (X, Y) pairs yielded a sample correlation coefficient of r = 0.02. (a) What is the conclusion that you would reach if you tested H0:p = 0 using a 0.05? What is...
The following data gave X = the water content of snow on April 1 and Y = the yield from April to July (in inches) on the Snake River watershed in Wyoming for 1919 to 1935. (The data were taken from...
A random sample of n = 25 observations was made on the time to failure of an electronic component and the temperature in the application environment in which the component was used. (a) Given that r...
Question 3 (24 marks) Wonderful Technology Company Limited sells computers and accessories. Data of the store's operations are as follow: Sales are budgeted at $400,000 for December 2019, $420,000...
Kratz Manufacturing Company uses an activity-based costing system. It has the following manufacturing activity areas, related cost drivers and cost allocation rates: Activity Cost Driver Cost...
You are a Partner with Fix-It Consultants and have been engaged in an advisory capacity with a software company, called MoveFast. The company is seeing a sharp decline in revenue, with the primary...

The Polar Decomposition: Prove that every invertible matrix A has a polar decomposition, written A = QB,

Question:

Step by Step Answer:

If A QB then K A T A B T Q T QB B 2 and hence B K ...View the full answer

Applied Linear Algebra

Students also viewed these Linear Algebra questions