Floquet's Theorem/Proof 2

Proof
Let $S\left({t}\right)=\Phi \left({t + T}\right){\Phi \left({T}\right)}^{-1}$, then we have that:

So $S\left({t}\right)$ is a fundamental matrix and $S\left({0}\right)=Id$ then $S\left({t}\right)= \Phi\left({t}\right)$ which it means that $\Phi \left({t + T}\right)=\Phi \left({t}\right)\Phi \left({T}\right)$.

Hence by the existence of the matrix logarithm, there exists a matrix $\mathbf B$ such that:
 * $\Phi \left({T}\right) = e^{\mathbf BT}$

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 a periodic function   with period $T$.

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