Show that if an n x n matrix A is positive definite, then there exists a positive

Question:

Show that if an n x n matrix A is positive definite, then there exists a positive definite matrix B such that A = B^TB. [Write A = PDP^T, with PT = P^-1. Produce a diagonal matrix C such that D = C^TC, and let B = PCP^T. Show that B works.

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

Step by Step Answer:

Related Book For book-img-for-question

Linear Algebra And Its Applications

ISBN: 9781292351216

6th Global Edition

Authors: David Lay, Steven Lay, Judi McDonald

See More Books

Question Posted: May 16, 2023 08:57 AM

See More Questions

An m n matrix A is said to have a right inverse if there exists an n x m matrix C such that AC = Im. A is said to have a left inverse if there exists an n m matrix D such that DA = In. (a) If A has...
An n x n symmetric matrix A is positive semi definite if xT Ax 0 for all x in Rn. Prove the following: (a) Every positive definite matrix is positive semi definite. (b) If A is singular and positive...
A nonnegative definite matrix A is positive definite if and only if it is nonsingular.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
The net accounts receivable on the books of GJY Corporation as of 1 January 20X3 are as follows: Accounts receivable ..........$ 562,000 Less: Allowance for sales discounts.... 12,000 Allowance for...
Grouper Corp. completed its first year of operations on December 31, 2022. Its initial income statement showed that Grouper Corp. had sales revenue of $235,818 and operating expenses of $98,853....
2. Suppose that r, ex are positive numbers, E is a convex subset of Rn such that E c Br(O), and there exists a sequence Xk E E such that Xk ---t 0 as k ---t 00. If f: Br(O) ---t R continuously...
Show that any second-order Markov process can be rewritten as a first-order Markov process with an augmented set of state variables. Can this always he done parsimoniously that is, without increasing...
Hasbro is considering dropping its Furby toy due to continuing losses. Data on the toy for the past year follow: Sales of 1 5 , 0 0 0 units $ 1 5 0 , 0 0 0 Variable expenses 1 2 0 , 0 0 0...
Robin travels between two cities and may use one of two routes: Route A is a fast four-lane highway, and route B is a long winding road. She has the habit of driving superfast. The highway patrol...
Suppose A is invertible and orthogonally diagonalizable. Explain why A 1 is also orthogonally diagonalizable.
Suppose A = PRP 1 , where P is orthogonal and R is upper triangular. Show that if A is symmetric, then R is symmetric and hence is actually a diagonal matrix.
Ed and Lita, age 68 and 65 respectively, are retired and collect the following income: a. What are Ed and Litas taxable Social Security benefits (SSBs)? b. If Litas pension is $25,000, what are Ed...
Consider the following double loop and identify the dependence. Verify the dependence(s) by applying GCD test and Bounds test. Use the direction vector of (1, -1) corresponding to T on (5M) S. L1: do...
After discovering a vulnerability in the passwd utility, the Linux developers have decided that it is too dangerous to continue to run the utility as root (through setuid). Unfortunately, there's no...
Your supervisor comes to you and says, that as a company, you're facing multiple communication challenges. Since transitioning to a remote workforce, communication has been extremely poor....
You are developing an application that will be running on several hundred Amazon EC2 instances. The application on each instance will be required to reach out through a file system protocol...
What is the danger of mixing elements ("hybrid strategy") from both agile and lean SC strategies?
Is the hedgehog signaling pathway a local or long-distance type of signaling? Explain.
True & False The basis of an asset must be reduced by the depreciation allowable, 2. Adjusted gross income (AGI) is the basis for a number of phase-outs of deductions. 3. A change to adjusted gross...
Fill in the blanks, choosing from: values of the response, values of the explanatory variable, observational units: In a scatterplot, the points are ________________; the x-axis represents...
The data shown in the following scatterplot show a very nice relationship between the two variables. However, the correlation here is 0.03, very close to zero. Explain why we can have a nice...
Which of the following is not a property of correlation, r? A. 1 r 1 B. Correlation measures the strength of a linear relationship between two quantitative variables. C. Th e sign on r tells the...
Mediocre Company has sales of $120,000, fixed expenses of $24,000, and a net income of $12,000. If sales rose 10%, the new net income would be: Question 18 options: $16,800 $36,000 $13,200 $15,600
1. Why might managers of small restaurants decide not to adopt the standard work hour approach to controlling labour cost? (minimum 150 words )
Which statement is true regarding the U.S. GAAP impairment test for limited life intangibles? A. U.S. GAAP impairment is likely to be greater than IFRS impairment. B. The impairment test for limited...

Show that if an n x n matrix A is positive definite, then there exists a positive

Question:

Step by Step Answer:

Let A PDP where P P The eigenvalues of A are all positive ...View the full answer

Linear Algebra And Its Applications

Students also viewed these Mathematics questions