Question: Let A be an n n matrix. Show that a vector x in Rn is an elgenvector belonging to A if and only if

Let A be an n × n matrix. Show that a vector x in Rn is an elgenvector belonging to A if and only if the subspace S of Rn spanned by x and Ax has dimension 1.

Step by Step Solution

★★★★★

3.44 Rating (170 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

The subspace spanned by x and A x will hav... 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)

949-M-L-A-E (777).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let A be an n ( n real matrix, all of whose eigenvalues are real. Prove that there exist an orthogonal matrix Q and an upper triangular matrix T such that QT AQ = T. This very useful result is known...

Let A be an n n matrix and let B = A + AT and C = A - AT (a) Show that B is symmetric and C is skew-symmetric. (b) Show that every n n matrix can be represented as a sum of a symmetric matrix and a...

Let A be an n n matrix and x Rn. (a) A scalar c can also be considered as a l l matrix C = (c), and a vector be Rn can be considered as an n 1 matrix B. Although the matrix multiplication CB is...

CO 250 Assignment 7 - Winter 2015 Due: Friday, March 13th 2015, by 4pm (sharp) Note: Submit your completed assignment to the following dropboxes outside MC 4066: Section 1 [L. Sanit`] a TTh...

Let A be an m x n matrix whose columns are linearly independent. a. Use Exercise 27 to show that A T A is an invertible matrix. b. Explain why A must have at least as many rows as columns. c....

Eigenvalue/Eigenvector Part IV: Suppose A is an n x n matrix. a. What does it mean to say that a vector x E Rn is an eigenvector of A? b. Suppose A E R is an eigenvalue of A. What is the eigenspace...

In a Hopfield neural network configured as an associative memory, with all of its weights trained and fixed, what three possible behaviours may occur over time in configuration space as the net...

9, DUE Friday, March 24th, 2017 Hand in Part A and Part B as two separate assignments. Include the following information in the top left corner of every assignment: your full name, instructor's last...

1. [6 marks] Determine whether the following function f on the convex set is convex, strictly convex, concave, strictly concave or neither: 1 1 f(x1, x2) = (x - 1) ex, N = {(x1, x2) R 2 : E 2. [17...

Let A be an n ( n matrix and ( a scalar. Show that the set W consisting of all vectors x in Rn such that Ax = (x is a subspace of Rn.

What are some of the main advantages and limitations of MRP?

5 Direct materials Direct labor Overhead (using plantwide rate) Total product cost Units produced Product cost per unit Activity Purchasing Depreciation of machinery Setup Total overhead Northwest's...

Artificial sweeteners. People who use artificial sweeteners in place of sugar tend to be heavier than people who use sugar. Does this mean that artificial sweeteners cause weight gain? Give a more...

2. Which of the following acts constitute investment according to the economists definition of that term? a. Pfizer builds a new factory in the United States to manufacture pharmaceuticals. b. You...

Let (p(x), q(x)) = p(0)q(0) + p(1)q(1) + p(2)q(2) in P2. In each case find the polynomial in U closest to f(x). (a) U = span{l + x, x2}, f(x) = 1 + x2 (b) U = span{l, 1 + x2}; f(x) = x

Using the inner product (p, q) = 10p(x)q(x) dx on P2, write v as the sum of a vector in U and a vector in UL. (a) v = x2, LJ = span{x + 1, 9x - 5} (b) v = x2 + 1, U = span{ 1, 2x - 1}

In each case, show that T is symmetric by calculating MB(T) for some orthonormal basis B. (a) T:R3 R3; T(a, b, c) = (a-2b, -2a + 2b + 2c, 2b-c); dot product (b) T:M22 M22; (c) T: P2 P2; T(a + bx +...

4. Assuming a risk-free rate of 8 percent and a market return of 12 percent, would a wise investor acquire a security with a beta of 1.5 and a rate of return of 14 percent given the facts above? 5....

Suppose there are two independent economic factors, M1 and M2. The risk-free rate is 7%, and all stocks have independent firmspecific components with a standard deviation of 55%. Portfolios A and B...

A $16,000 loan is to be amortized for 10 years with quarterly payments of $610.84. If the interest rate is 9%, compounded quarterly, what is the unpaid balance immediately after the sixth payment?...