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
Algebra and Trigonometry (6th Edition)
6th Edition
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Author:Robert F. Blitzer
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7b32e38-f38f-485d-a004-13b874e267a1%2F44aba604-82d4-4a4f-9c98-daf53e20c62a%2F3znqcdn.jpeg&w=3840&q=75)
Transcribed Image Text: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
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