Problem 10. Suppose that x is an eigenvector for a matrix A with eigenvalue A = -2.Compute A5x.(a) -10x(b) -32x(c) 3x(d)C-2x(e) None of the aboveDELL

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Answered: Problem 10. Suppose that x is an…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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If matrix A is square of size n x n with a rank equal to 2 and n > 2. The rank of a matrix is equal to the number of non-zero eigenvalues. Can you write A as the product of two matrices B E Rnxq and C E RPX" where we require that p < n and q < n? You may assume that A is diagonalizable. Show how to construct a matrix B and C such that A = BC.
A 2x2 upper triangular matrix A has eigenvalues a and - 5 (with a ≠ -5). An eigenvector corresponding to the eigenvalue a is (1,0). Which of the following could be the matrix A? A. (-5 a/2 0) B. (a 0/2 -5) C. (-5 2/0 a) D. (a -5/0 2) E. (a 2/0 -5)
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Suppose A is a 3 x 3 matrix with eigenvectors [1] V1 = V2 = 1 and v3 = corresponding to eigenvalues A1 =-, d2 = }, and A3 = 1, respectively. Find A and A20.
4. A real, symmetric 3-by-3 matrix M has eigenvalues X₁ = 4, X₂ 1 vector corresponding to the eigenvalue X₁ = 4 is v₁ -1 0 [4] -2 to the eigenvalue X₂ = 6 is v₂ = 6 and X3 = -3. An eigen- and an eigenvector corresponding What is M?
Suppose that you were trying to use the Putzer method to exponentiate the matrix A which has eigenvalues dı = 4 and A2 = 6, in that order. ao(t) is always the function et. Choose the correct equation for a1(t) da1 О (а) 6a1 + e4t a1(0) = 1 dt da1 : 4a1 + e4t P (g) da1 a1(0) = 1 (c) dt -4a1 + ett a1(0) = 0 da1 (d) dt 4a1 + et a1(0) = 0 da1 (e) : 6a1 + e4t a1(0) = 0 dt da1 (f) dt — 4а, + e# a,(0) — 1
4. Consider the matrix equation Ax = b where, 4- (19) A= = where a, b > 0. (a) When is A strictly diagonally dominant and what does this tell us about the Gauss- Seidel method for Ax : b. = (b) Find the iteration matrix for the Gauss-Seidel method applied to Ax = b. (c) Using your previous answers, for what values of a and b does the Gauss-Seidel method converge.

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Problem 10. Suppose that x is an eigenvector for a matrix A with eigenvalue λ = -2. Compute A5x. (a) - 10x (b) -32x (c) 3x (d) - 2x (e) None of the above