11. Which of the following is a least square solution of the over-determined linear system-1-2(a)(b) (7)(c)(d) ()(e) None of the above9. Which of the following is the eigenspace for the eigenvalue 3 of the matrix ( ):4.{()}(a) span(b) span(c) R?(d) {0}(e) None of the above10. Which of the following matrices is NOT diagonalizable?(a)(b) 6(0) (: 3)(d)-1(e) None of the above - they are all diagonalizable

Answered: Image /qna-images/answer/6bd63f33-8bd4-450b-bc2f-c180f0f8d0b4.jpg

Answered: 11. Which of the following is a least…

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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For a linear DDS with multiple equations, assume that A is the matrix of the system with eigenvalue c and eigenvector u. Which of the following is true? Check all that apply. X(n) = A"X(0) is the recursive equation for the DDS □X(n) = A"X(0) is the explicit solution for the DDS □X(n) = AX(n − 1) is the recursive equation for the DDS We can always compute X(n) as X(n) = c"X(0) using the eigenvalue c We can compute X(n) as X(n) = X(0) when X(0) is a multiple of
31. (a) Let A be an n × n-symmetric matrix. Diagonalize A to show that x • Ax 2 || x || ² is greater than or equal to the smallest eigenvalue of A for all x # 0 in R". (b) Show that the quadratic form Q₁(x) = x Ax is coercive if and only if A is positive definite. (c) Conclude from (b) that if f(x) = a + b x + 1x. Ax is any quadratic function where a = R, be R₁ and A is an n - n-symmetric matrix, then f(x) is coercive if and only if A is positive definite.
24. (T/F) The matrices A and B-¹AB have the same sets of eigenvalues for every invertible matrix B.
5. Let A be a 6 x 6 matrix with characteristic equation X2(a – 1) (A – 2)3= 0. What are the possible dimensions for eigenspaces of A?
Use (a) to solve the system x = (: :) . Х. -8 7 NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly independent solutions. The second will yield solutions which are identical (up to a constant) to the solutions already found. a) is as follows Show from first principles, i.e., by using the definition of linear independence, that if x+iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then the two real solutions Y(t) = et (u cos bt – w sin bt) and Z(t) = eat (u sin bt + w cos bt) are linearly independent solutions of X = AX.
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
solve for iv with some detail

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