Unformatted text preview: the Floquet exponents are well defined. However, it is perfectly possible for a Floquet multiplier
to be negative, in this case the corresponding exponent would be pure imaginary. Floquet theory will be concerned with the logarithm of the monodromy matrix. We next define
the matrix logarithm. Begin with a preliminary remark.
The matrix exponential
was defined by a series expansion patterned after the Maclaurin series
. A similar procedure will be used to define the logarithm of a nilpotent
, converging for
. If we now integrate both
sides and evaluate the constant of integration, we find
obtain . Now replace , also
, where is a nilpotent matrix, to .  There are no convergence issues because
Lemma 2.12 Any nonsingular matrix is nilpotent!
has a (possibly complex) logarithm .
is the semisimple-nilpotent decomposition,
eigenvalues of repeated according to multiplicity, is the maximum algebraic multiplicity
of any eigenvalue and is the matrix of generalized eige...
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- Fall '14
- Linear Algebra, Complex number, Floquet Theory