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Let A E L(R). Suppose all solutions of x' = Ax are periodic with the same period. Then A is semisimple and the characteristic polynomial is a power of t² + a², a € R.

Let A E L(R). Suppose all solutions of x' = Ax are periodic with the same period. Then A is semisimple and the characteristic polynomial is a power of t² + a², a € R.

Advanced Engineering Mathematics
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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2. Let AL(R"). Suppose all solutions of x' = Ax are periodic with the same period. Then A is semisimple and the characteristic polynomial is a power of t² + a², a € R. 2. If x is an eigenvector belonging to an eigenvalue with nonzero real part, then the solution etax is not periodic. If ib, ic are pure imaginary eigenvalues, b ‡ ±c, and z, w E C" are corresponding eigenvectors, then the real part of e¹4 (z + w) is a nonperiodic solution.
Transcribed Image Text:2. Let AL(R"). Suppose all solutions of x' = Ax are periodic with the same period. Then A is semisimple and the characteristic polynomial is a power of t² + a², a € R. 2. If x is an eigenvector belonging to an eigenvalue with nonzero real part, then the solution etax is not periodic. If ib, ic are pure imaginary eigenvalues, b ‡ ±c, and z, w E C" are corresponding eigenvectors, then the real part of e¹4 (z + w) is a nonperiodic solution.
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