# Then of 2 with 14 chapter 2 part b where is the

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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 for where . A similar procedure will be used to define the logarithm of a nilpotent matrix. Recall , converging for . If we now integrate both sides and evaluate the constant of integration, we find converging for obtain . Now replace , also , where is a nilpotent matrix, to . [4] There are no convergence issues because Lemma 2.12 Any nonsingular matrix is nilpotent! has a (possibly complex) logarithm . Here, is the semisimple-nilpotent decomposition, with 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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