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formed. Example. Find the logarithm of
The roots of the characteristic equation give . Then recalling Euler’s formula,
and Write to find and and solve for the . The transformation matrices are
.■ . Finally Example. Find the logarithm of and, . Since is upper triangular, the eigenvalues are on the main diagonal. has 1 eigenvalue,
, with multiplicity
The associated generalized eigenspace is all of
simplest choice for a basis is
. Then the transformation matrices
. We thus have
has a nilpotent part so we have another term in the logarithm to compute. Note . 16
Chapter 2 part B Let . Then . According to  .■ Recall that we are trying to characterize the solutions
of the initial value problem for a periodic linear system . The fundamental matrix solution
is the solution of the matrix
initial value problem . Any solution of  can be written
monodromy matrix is the solution of , with
, after one period:
Theorem 2.13 (Floquet). Let be the monodromy matrix for a -periodi...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
- Fall '14