Question: Let K > 0 be a positive definite n x n matrix. Prove that an n n matrix S satisfies ST K S =

Let K > 0 be a positive definite n x n matrix. Prove that an n × n matrix S satisfies ST K S = I if and only if the columns of S form an orthonormal basis of Rn with respect to the inner product (v, w) = vT K w.

Step by Step Solution

★★★★★

3.29 Rating (167 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

If S v 1 v 2 v n then the i j e... View full answer

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)

952-M-L-A-E (2245).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

(a) Prove that if K1, K2 are positive definite n x n matrices, then K Is a positive definite 2n Ã 2 n matrix. (b) is the converse true? Ki O (O

Let K > 0 be a positive definite matrix. Characterize the matrix norm induced by the inner product (x. y) = xT K y. Use Exercise 8.4.45.

Let S be a positive definite n n matrix. For any x in Rn define ||x|| = (xtSx)1/2. Show that this defines a norm on Rn. [Use the Cholesky factorization of S to show that xtSy = ytSx ...

2.3 Consider the system x' = f(t, x), x (to) = xo, to 0, where the function f : [0, ) x R" R" is continuous in (t, x), locally Lipschitz in x and satisfies the condition (f(t, x), Px) 0, \t0, x R",...

Let A be a symmetric positive definite n x n matrix with entries a;; that are nonzero only if one of the following holds: i = 1, or i = n, or j = 1, or j = n, or i = j. Otherwise, aj = 0. (a) Show...

20. Let R be a positive definite n x n matrix and A be a symmetric n x n matrix. Show that, for some positive value of the real number , the matrix B is a positive definite matrix, where B = R - A.

A positive definite n x n matrix A can be factored into the form LDL ^ T ( where ^ T means transpose " ) using the following algorithm: 1 : Input: matrix A and its size n 2 : for i = 1 to n do 3 :...

1. Let B = (v1, .... Un), C= {ul,....un}, and D = (w1,..., Win} be three bases for R". (a) Prove the following identity regarding the change of basis matrices: Pc +DPBAC = l'B +D. (b) Consider the...

,.. I need your input here please Amanda's utility function is Leontief and given by Wa(Xarya) = min(xa. yal while Brenda's utility is linear and given by us(xb. )b) = XD + VD Their endowments are wa...

,, kindly solve Consider the Linear Regression Model 1. For any X = x, let Y = x8 + U, where B E R*. 2. X is exogenous. 3. The probability model is { f (u; ) is a distribution on R : E, [U] = 0, Var,...

XYZ Corporation, which purchases outdoor clothing and equipment and resells them, began business as a merchandising corporation on January 1, 2022. At the end of January, the company prepared the...

Add additional information to the student class below such as name, address, major, and contact information along with all the getters and setters (methods) needed to access all data. Submit your...

A study was done by randomly assigning 200 volunteers with sore throats to either drink a cup of herbal tea or use a throat lozenge to ease their pain. The percentages reporting relief for the two...

3. Suppose that the two nations in Problems 1 and 2 choose to specialize in producing the goods for which they have a comparative advantage. They agree to trade at a rate of exchange of

- 44.1 / (-6.3)

4 cubed + 10 X 20 + 8 squared 23

-6m + 24

Bowen Company manufactures one product, it does not maintain any beginning or ending inventories, and its uses a standard cost system. Its predetermined overhead rate includes $1,000,000 of fixed...

Question 2. (10 Marks in total). Let Y, I, and E be random variables such that E = V/8 and r = if I > 0, otherwise , where I is uniformly distributed on the interval (-1, 1). What is the probability...

Assume the current U S dollar-yen spot rate is 125 Further. the current nominal 180-day rate of return in Japan is 3% and 4% n the at is the approximate forward exchange rate for 180 days? 123 8O 124...