Question: (a) Let K and L be symmetric n x n matrices. Prove that xTKx = xTx for all x Rn if and only if

(a) Let K and L be symmetric n x n matrices. Prove that xTKx = xTx for all x ∈ Rn if and only if K = L.
(b) Find an example of two non-sy mmetric matrices K ≠ L such that xTKx = xTLx for all
X € Rn.

Step by Step Solution

★★★★★

3.23 Rating (161 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a First k ii e T i Ke i e T i Le i l ii so their diagon... 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 (2029).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let A be a symmetric n Ã n matrix with eigenvalues Î»1,..., Î»n. Show that there exists an orthonormal set of vectors {x1, ..., xn} such that

Let A be a symmetric n Ã n matrix with orthonormal eigenvectors u1 . . . . un. Let x Rn and let ci = uTix for i = 1,2,. . . .,n show that (a) (b) if x 0, then (c) AxlI2 [Alla max .

Let AT = -A be a real, skew-symmetric n n matrix. (a) Prove that the only possible real eigenvalue of A is = 0. (b) More generally, prove that all eigenvalues of A are purely imaginary, i.e., Re ...

1) {1 point) Suppose that A = [42 34] = LDU where: 1 G L = [ 1] is a lower triangular matrix with ones on the diagonal, a. [la I1] _ _ _ D = IS a diagonal matrix, and [l c 1 d U = [ 1] is an upper...

1. [-/1 Points] DETAILS 0/100 Submissions Used 1 2 0 0 Solve the equation for X , given that A = and B = 3 4 -1 1 2(A + 2B) = 3X X = Need Help? Read It Submit Answer2. [-/1 Points] DETAILS 0/100...

Denote by Sax the vector space of symmetric n x n matrices and observe that ATA S*** VA R***, where the superscript T denotes transpose. So, define f: Rxn Saxn by f(A) := ATA. (a) Explain why f is...

Problem #5: Let A and B be n x n matrices. Which of the following statements are always true? (i) If det(A) = det(B) then det(A B) =0. (ii) If A and B are symmetric, then the matrix AB is also...

Let F2 = {0,1} with the usual arithmetic modulo 2. The set Fy = {0,1}" consisting of all n-bit vectors (vectors of length n with entries 0 or 1) is an n-dimensional vector space over F2, with the...

please help me solve this. 1. (12 pts) Let K and M be two m x m symmetric matrices. a Prove that matrix KM is not necessarily symmetric (To prove it, you may use prop- erties or provide a...

Anns electric clothes dryer was unreliable, and she sought a replacement. She noticed an advertisement in the newspaper for a used dryer, in like-new condition, at a price of $200. She visited the...

Two fungal diseases caused in animals are Options: 1) Ringworm, Facial Eczema 2) Foot and mouth disease, Rabies 3) Vibriosis, Pinkeye 4) Brucellosis, Strangles

Why will scarcity continue to be a problem in the future? A Prices will rise B The quantity of resources will decline C Wants will continue to increase D World population will fall

What caused Mr. Stevensons anguish? Was it the correction process?

Let A = Q1R1 = Q2R2, where Q1 and Q2 are orthogonal and R1 and R2 are both upper triangular and nonsingular. (a) Show that QT1Q2 is diagonal. (b) How do R1 and R2 compare? Explain.

Let A = xyT, where x Rm, y Rn, and both x and y are nonzero vectors. Show that A has a singular value decomposition of the form H1H2, where H1 and H2 are Householder transformations and 1 = ||x||...

In constructing a Householder matrix to zero out the last n - l of a vector x in Rn, the scalar a must satisfy || = ||x||2. Instead of always taking a = ||x||2. it is customary to choose a = -||x||2...

To estimate the amount of lumber in a tract of timber, an owner randomly selected seventy 50-by50foot squares, and counted the number of trees with diameters exceeding 12 inches in each square. The...

A 4-year project, X, with an upfront cost of $20 million is expected to yield the following series of cash flows: $3.60 million in year one, $4.20 million in year two, $5.70 million in year three,...

Question 4 of 6 For 5 years, Janet saved $600 at the beginning of every month in a fund that earned 4.5% compounded annually. a. What was the balance in the fund at the end of the period? Round to...