Question: Prove that if K is any positive definite matrix, then any positive scalar multiple c K, c > 0, is also positive definite.

Prove that if K is any positive definite matrix, then any positive scalar multiple c K, c > 0, is also positive definite.

Step by Step Solution

★★★★★

3.29 Rating (170 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

First cK T cK T cK ... View full answer

blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Document Format (1 attachment)

Word file Icon

952-M-L-A-E (2049).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Q:

Prove that if K is a positive semi-definite matrix, and f rng K. then the quadratic function p(x) = xT K x - 2xT f + c has no minimum value.

Q:

Let A" be a nonsingular symmetric matrix. (a) Show that xTK~1x = yTK y, where Ky = x. (b) Prove that if K is positive definite, so is K-1.

Q:

Prove that if K is symmetric, diagonally dominant. and each diagonal entry is positive, then K is positive definite.

Q:

(a) Prove that if K is a positive definite matrix, then K2 is also positive definite. (b) More generally, prove that if S = ST is symmetric and nonsingular, then S2 is positive definite.

Q:

net,,,,,, 8 Computation Theory (a) Define precisely what is meant by the following: (i) f(x1, x2, . . . xn) is a Primitive Recursive (PR) function of arity n. [5 marks] (ii) f(x1, x2, . . . xn) is a...

Q:

Given a vector space V over the field R, an Inner Producton V is a (bilinear) function with the following properties 1 . Symmetry-(x, y)-(xx) 2. Positive definiteness-(x, x? 0 & ?x?x)-0-x-0 3....

Q:

(a) Prove that a positive definite matrix has positive determinant: det K > 0. (b) Show that a positive definite matrix has positive trace: tr K >0. (c) Show that every 2 x 2 symmetric matrix with...

Q:

24. Let A be a k x k positive definite matrix, a be a k x 1 vector, and be a scalar such that > a'A-1a; show that A* is a positive definite matrix when A* is defined by A* = [ A a ] [ a' ]

Q:

24. Let A be a k x k positive definite matrix, a be a k x 1 vector, and be a scalar such that > a'A-1a; show that A* is a positive definite matrix when A* is defined by A* = [ A a ] [ a' ]

Q:

Prove Proposition 9.17. A particularly important class of systems are the linear gradient flows in which AT is a symmetric, positive definite matrix. According to Theorem 8.23, all the eigenvalues of...

Q:

Determine the magnitude of the hydrostatic force acting per foot of length on the seawall. ? w = 62.4 lb/ft 3 . y = -2x 8 ft 2 ft

Q:

How would you proceed to estimate empirically the parameter "" in the Huff retail location model for a branch bank?

Q:

NJM Incorporated is a football manufacturer.NJM has fixed operating costs of $ 4 0 0 , 0 0 0 and variable costs of $ 1 2 per football.The footballs sell for $ 3 5 each and NJM plans to sell 3 0 0 , 0...

Q:

Name the two conflicting public policy concerns the court has about requiring disclosure vs. not requiring disclosure.

Q:

Prove: If A and B is m à n matrices, then A and B are row equivalent if and only if A and B have the same reduced row-echelon form. 524 021 101 319 241

Q:

Suppose that A is some unknown invertible matrix, but you know of a sequence of elementary row operations that produces the identity matrix when applied in succession to A. Explain how you can use...

Q:

Indicate whether the statement is always true or sometimes false. Justify your answer with a logical argument or a counterexample. (a) If A is a singular n n matrix, then Ax = 0 has infinitely many...

Q:

Question 1 0.3 pts A manager should never pursue a project with a positive net present value (NPV > 0). True False Question 2 0.3 pts The Opportunity Cost of Capital (another name for the discount...

Q:

BOTH ANSWERS PLS (Related to Checkpoint 6.1) (Future value of an annuity) Imagine that Homer Simpson actually invested the $150,000 he earned providing Mr. Burns entertainment 6 years ago at 8...

Q:

TBurch Inc., is considering a new three-year expansion project that requires an initial fixed asset investment of $2.4 million. The fixed asset will be depreciated straight-line to zero over its...