Floquet's Theorem/Proof 1

Proof
We assume the two hypotheses of the theorem.

We have that:

So the first implication of the theorem holds, that is: that $\map \Phi {t + T}$ is a fundamental matrix.

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

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

Defining $\map {\mathbf P} t = \map \Phi t e^{-\mathbf B t}$, it follows that:

and hence $\map {\mathbf P} t$ is periodic with period $T$.

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