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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Transcribed Image Text:1. Let A be a 2 × 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-(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.
1 4
has 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 that
J = P-'AP = (
-3 1
0 -3
(d) Hence find a fundamental set of solutions r(t), x(2)(t) and a fundamental
matrix 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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