1. Let A be a 2 × 2 matrix with a double eigenvalue r, and let T be an invertiblematrix such that T-'AT =Let I be the identity matrix.(a) Show that T-(A – rI)THence show that (T-(A – rI)T)² = 0.Deduce that (A – rI)² = 0.(b) Now let ŋ be a vector such that { = (A-rI)n+0. Show that { is an eigenvectorof A.1 4has a double eigenvalue A = -3. Let n=6):()-(c) The matrix A =Compute { = (A – AI)ŋ and show that { + 0. Define P = (§,n) and show thatJ = P-'AP = (-3 10 -3(d) Hence find a fundamental set of solutions r(t), x(2)(t) and a fundamentalmatrix F(t) for the system r = Ar.(e) Find eAt by first computing eJt and using that J = P-'AP.(f) Find (F(0))-! and verify that e4t = F(t)(F(0))-!.

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Let A be a 2 x 2 matrix with a double eigenvalue r, and let T be an invertible matrix such that T-'AT = (" :). Let I be the identity matrix. (a) Show that T-'(A – rI)T = (% ): Hence show that (T-1(A – rI)T)² = 0. Deduce that (A – rI)² = 0. (b) Now let ŋ be a vector such that { = (A-rI)n#0. Show that { is an eigenvector of A. (c) The matrix A = G) has a double eigenvalue A = -3. Let ŋ = (6). Compute § = (A – AI)ŋ and show that { + 0. Define P = (§, n) and show that J = P-'AP = %3D -3 1 -3

Let A be a 2 x 2 matrix with a double eigenvalue r, and let T be an invertible matrix such that T-'AT = (" :). Let I be the identity matrix. (a) Show that T-'(A – rI)T = (% ): Hence show that (T-1(A – rI)T)² = 0. Deduce that (A – rI)² = 0. (b) Now let ŋ be a vector such that { = (A-rI)n#0. Show that { is an eigenvector of A. (c) The matrix A = G) has a double eigenvalue A = -3. Let ŋ = (6). Compute § = (A – AI)ŋ and show that { + 0. Define P = (§, n) and show that J = P-'AP = %3D -3 1 -3

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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