Let X' =AX be a system of n lincar differential equations where X is an n- tuple of differentiable functions x,(t), x2(t), ... , x,(t) of the real variable t, and A is ån n x n coefficient matrix as in Exercise 14 of Section 5.2. In contrast to that exercise, however, suppose that A is not diagonalizable, but that the characteristic polynomial of A splits. Let 1,, A2,..., be the distinct eigenvalues of A. (a) Prove that if u is the end vector of a cycle of generalized eigenvectors of L, of length p and u corresponds to the eigenvalue A, then for any polynomial f(t) of degree less than p the function ed«[S(t{A – 2,1)° -+ f"(9(A – 2,1)P -2 + ... + f®-()]u is a solution to the system X'= AX. (b) Prove that the general solution to X' = AX is a sum of functions of the form given in part (a), where the vectors u are the end vectors of the distinct cycles that constitute a fixed Jordan canonical basis for L

Let X' =AX be a system of n lincar differential equations where X is an n- tuple of differentiable functions x,(t), x2(t), ... , x,(t) of the real variable t, and A is ån n x n coefficient matrix as in Exercise 14 of Section 5.2. In contrast to that exercise, however, suppose that A is not diagonalizable, but that the characteristic polynomial of A splits. Let 1,, A2,..., be the distinct eigenvalues of A. (a) Prove that if u is the end vector of a cycle of generalized eigenvectors of L, of length p and u corresponds to the eigenvalue A, then for any polynomial f(t) of degree less than p the function ed«[S(t{A – 2,1)° -+ f"(9(A – 2,1)P -2 + ... + f®-()]u is a solution to the system X'= AX. (b) Prove that the general solution to X' = AX is a sum of functions of the form given in part (a), where the vectors u are the end vectors of the distinct cycles that constitute a fixed Jordan canonical basis for L

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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