Floquet's Theorem

Theorem
If: Then:
 * $A(t)$ is a continuous matrix function with period $\omega$, and
 * $\Phi(t)$ is a fundamental matrix of the Floquet system $x' = A(t)x$,
 * $\Phi(t+\omega)$ is also a fundamental matrix; moreover
 * There exists:
 * a nonsingular, continuously differentiable matrix function $P(t)$ with period $\omega$ and
 * a constant (possibly complex) matrix $B$ such that
 * $\Phi(t) = P(t) e^{Bt} $.

Proof
Allow the two hypotheses of the theorem by assumption.

Since
 * $\frac{d}{dt} (\Phi(t+\omega)) = \Phi'(t+\omega) = A(t+\omega) \Phi(t+\omega) = A(t) \Phi(t+\omega)$,

the first implication of the theorem obtains.

Because $\Phi(t)$ and $\Phi(t+\omega)$ are both fundamental matrices, there must exist some $C$ such that
 * $\Phi(t+\omega) = \Phi(t)C$,

hence by the existence of the matrix logarithm, there exists a $B$ such that
 * $C = e^{B\omega}$.

Defining $P(t) = \Phi(t) e^{-Bt}$, it follows that
 * $P(t+\omega) = \Phi(t+\omega) e^{-Bt-B\omega}$
 * $=\Phi(t)C e^{-B\omega} e^{-Bt} $
 * $=\Phi(t) e^{-Bt}$
 * $=P(t)$

and hence $P(t)$ is periodic with period $\omega$.

As $\Phi(t) = P(t) e^{Bt}$, the second implication also obtains.