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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 $(1)$?

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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.

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  • $\begingroup$ Thank you very much. Now I understand it. $\endgroup$ – Fred Oct 22 '18 at 16:07

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