Eigenvalues and eigenvectors for time-periodic differential operator

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I'm trying to solve and simplify a bigger equation, for this purpose I want to find the eigenvalues and eigenvectors of this differential operator: $$ ( A(t) - \frac{d}{dt} ) $$ where A(t) is T-periodic matrix

I can't find a way to find the eigenvalues $q_\beta$ and eigenvectors $v_\beta(t)$ of the equation $$ ( A(t) - \frac{d}{dt} ) v_\beta = q_\beta v_\beta $$

my guess is that $v_\beta$ is also $T$-periodic column vectors.

What do you think?