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Unformatted text preview: c linear system
its logarithm. Then there exists a -periodic matrix such that the
fundamental solution is
Proof. Give as in the text.
Remark. Note that
period the matrices
). and . This implies
for an integer. Then
. However, when is not an integer multiple of a
may be complex (consider
is the square root of Alternatively there is a real form of Floquet’s theorem. It is based upon the fact that the square
of any real matrix has a real logarithm (Exercise #21).
Theorem 2.14. Let be the fundamental matrix solution for the time T-periodic linear system
. Then there exists a real -periodic matrix and a real matrix such that
Proof. From exercise 21, for any nonsingular matrix
, and then there is a real matrix .
Therefore, is -periodic. □ such that...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
- Fall '14