Floquet's Theorem/Proof 2

Proof
Let $\map S t = \map \Phi {t + T} {\map \Phi T}^{-1}$.

Then:

So $\map S t$ is a fundamental matrix and:
 * $\map S 0 = Id$

Then:
 * $\map S t = \map \Phi t$

which means that:
 * $\map \Phi {t + T} = \map \Phi t \map \Phi T$

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

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

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

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