Let A be a positive definite symmetric matrix. Show that there exists an invertible matrix B such

Question:

Let A be a positive definite symmetric matrix. Show that there exists an invertible matrix B such that A = BT B. [Hint: Use the Spectral Theorem to write A = QDQT. Then show that D can be factored as CT C for some invertible matrix C.]

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

Step by Step Answer:

Related Book For book-img-for-question

Linear Algebra A Modern Introduction

ISBN: 978-1285463247

4th edition

Authors: David Poole

See More Books

Question Posted: Apr 15, 2016 06:23 AM

See More Questions

Let A be a positive definite symmetric matrix. Show that there is a positive definite symmetric matrix B such that A = B2. (Such a matrix B is called a square root of A.)
(a) Show that, for a positive definite, symmetric matrix A, Theorem 7 .13 gives the orthogonal diagonalization of A, as guaranteed by the Spectral Theorem. (b) Show that, for a positive definite,...
Let C = [Cij] be an n x n positive definite symmetric matrix and let V be an n-dimensional vector space with ordered basis S = {u1, u2,... un}. For v = a1u1+ a2u2+...+anun and w = b1u1+b2u2+ ... +...
Economic crises tend to occur sporadically virtually every decade and in various countries ranging from Sweden to Argentina, from Russia to Korea, and from Japan to the United States. Each crisis is...
The post-closing trial balance of Dondec Corporation at December 31, 2014, contains these stockholders equity accounts. Preferred Stock (6,000 shares issued).................$ 300,000 Common Stock...
The following selected transactions were completed by Padre Co. during October of the current year: Oct. 1. Purchased merchandise from Wood Co., $15,500, terms FOB shipping point, 2/10, n/eom....
Can a duty to care be established by statute or contract? Discuss your answer.
Confucius Bookstores inventory is destroyed by a fire on September 5, 2015. The following data for year 2015 are available from the accounting records. Estimate the cost of the inventory destroyed....
Your company is excited with the information discussed regarding the cloud technologies that are available to Get-Sole Shoes. Now the management team wants to examine what its cloud infrastructure...
Dan Summerfield just recently took over as director of purchasing for Great Western University. Great Western spent roughly $400,000 a year for the purchase of various kinds of plumbing supplies....
Let B be an invertible matrix. Show that A = BT B is positive definite.
Let A and B be positive definite symmetric n ( n matrices and let c be a positive scalar. Show that the following matrices are positive definite. (a) cA (b) A2 (c) A + B (d) A-1 (First show that A is...
4/9 2 Perform the indicated operation by hand.
Can knowing the source of information that you find online unfairly bias you against the information? Explain your answer.
What is visual literacy?
Read the following rsum information and then (1) analyze the strengths or weaknesses of the information and (2) revise the rsum so that it follows the guidelines presented in this chapter. William...
According to the World Cancer Research Fund (www.wcrf.org), in 2018 the age-standardized rate for all cancers was 197.9 per 100,000 people. Australia has the dubious honor of having the highest...
Your task is to deliver a fiveminute presentation, without visuals, to your class, with the aim of persuading them to donate 1 (or its equivalent value in your currency) to the charity of your...
The only true way for balance sheets to be meaningful to users would be for all financial assets and liabilities to be measured at fair value with gains and losses recognised in profit or loss....
(8%) Problem 6: A student attaches a f= 3.5 kHz oscillator to one end of a metal rail of length L = 25 m. The student turns on the oscillator and uses a piezoelectric gauge at the other end to...
Find a basis for the span of the given vectors. 2 1 1 1 2
Find a basis for the span of the given vectors. [2 -3 1], [1 -1 0], [4 -4 1]
Find a basis for the span of the given vectors. [3 1 -1 0], [0 -1 2 -1], [4 3 8 3]
According to Larry Swedroe (in his talk & in his book), why might an investor expect investments in companies with high ESG scores ("green companies") to underperform investments in companies with...
A paper machine makes frequent weight changes. The production management feels that the stock flow can be safely moved at a maximum rate of 50 gpm per minute. Before trying this on the machine they...
State of Economy Probability of State 31.1 Consider the following information on a portfolio of three stocks: Rate of Return in Percent Stock A Stock B Stock C Occurring Portfolio Weights Boom Normal...

Let A be a positive definite symmetric matrix. Show that there exists an invertible matrix B such

Question:

Step by Step Answer:

Since A is symmetric and positive definite we can find an orthogonal m...View the full answer

Linear Algebra A Modern Introduction

Students also viewed these Linear Algebra questions