MATH UN1102 Chapter Notes - Chapter 12: Generalized Eigenvector

• School: Columbia University
• Department: Mathematics
• Course: MATH UN1102
• Professor: Nathan Dowlin

Document Summary

Using the same procedure as in hw 11, the eigenvalues of the matrix are r = 1 2i and the eigenvectors are = (cid:18) 2i. 1 (cid:19) e 1t(cos 2t + i sin 2t) + c2(cid:18) 2i. 1 cos 2t (cid:19) + ie t(cid:18) 2 cos 2t x = c1(cid:18) 2i. 1 (cid:19) e 1t(cos 2t i sin 2t) sin 2t (cid:19)(cid:19) + c2(cid:18)e t(cid:18) 2 sin 2t cos 2t (cid:19) + ie t(cid:18) 2 cos 2t. Sin 2t (cid:19) cos 2t (cid:19) + (c1 c2)ie t(cid:18) 2 cos 2t cos 2t (cid:19) + d2e t(cid:18) 2 cos 2t. Using the same procedure as in hw 11, the eigenvalues of the matrix are r = 1 i and the eigenvectors are = (cid:18) 1. 2 + i (cid:19) thus a solution is (cid:18) 1. 2 i (cid:19) e( 1+i)t. separating the real and imaginary parts, yields: 2 i (cid:19) e( 1+i)t = (cid:18) 1 (cid:18) 1.