Floquet's Theorem

Theorem
Let $\mathbf A \left({t}\right)$ be a continuous matrix function with period $T$.

Let $\Phi \left({t}\right)$ be a fundamental matrix of the Floquet system $\mathbf x' = \mathbf A \left({t}\right) \mathbf x$.

Then $\Phi \left({t + T}\right)$ is also a fundamental matrix.

Moreover, there exists:
 * A nonsingular, continuously differentiable matrix function $\mathbf P \left({t}\right)$ with period $T$
 * A constant (possibly complex) matrix $\mathbf B$ such that:
 * $\Phi \left({t}\right) = \mathbf P \left({t}\right) e^{\mathbf Bt}$

Proof
We assume the two hypotheses of the theorem.

We have that:

So the first implication of the theorem holds, i.e. that $\Phi \left({t + T}\right)$ is a fundamental matrix.

Because $\Phi \left({t}\right)$ and $\Phi \left({t + T}\right)$ are both fundamental matrices, there must exist some matrix $\mathbf C$ such that:
 * $\Phi \left({t + T}\right) = \Phi \left({t}\right) \mathbf C$

Hence by the existence of the matrix logarithm, there exists a matrix $\mathbf B$ such that:
 * $\mathbf C = e^{\mathbf BT}$

Defining $\mathbf P \left({t}\right) = \Phi \left({t}\right) e^{-\mathbf B t}$, it follows that:

and hence $\mathbf P \left({t}\right)$ is periodic with period $T$.

As $\Phi \left({t}\right) = \mathbf P \left({t}\right) e^{\mathbf B t}$, the second implication also holds.