# Floquet theorem in mathematics and physics

I am using the notation from Proofwiki. The Floquet theorem states that for a continuous matrix function $$A(t)$$ with period $$T$$ and a fundamental matrix $$\Phi(t)$$ of the system $$x'(t)=A(t)x(t)$$, it is possible to write

$$(1)\quad\Phi(t)=P(t)e^{Bt}$$

with $$T$$-periodic, nonsingular, continuously differentiable matrix function $$P(t)$$ and constant, possibly complex matrix $$B$$.

On the other hand, in many physics papers, e.g. eq. 5.11 of this source, it is stated (with different notation) that

$$(2)\quad\Phi_\alpha(t)=e^{-i\epsilon_\alpha \frac t\hbar}P_\alpha(t)$$

with $$\Phi_\alpha$$ ($$P_\alpha$$) being a column vector of $$\Phi$$ ($$P$$) and $$\epsilon_\alpha$$ a scalar.

However, I have not been able to find a proof for $$(2)$$. To me it seems like a stronger statement, since it needs $$B$$ to be a diagonal matrix as the factors $$i\epsilon_\alpha/\hbar$$ are only scalars.

Thus, my question regards this lack of proof. Can someone provide me with a proof or at least a criterion for which the $$(2)$$ follows from $$(1)$$?

In physics, $$\Phi, P$$ are (in most applications) unitary matrices and $$B$$ is an anti-Hermitian matrix. Because of this, you can diagonalize the statement (1) [i.e., calculate the eigenvalues and eigenvectors of $$B$$] and arrive at (2) for the eigenvectors.