If we have a system of equations of the form
$$\left(\begin{array}{c} f_1(t)\\ f_2(t)\\ \vdots\\ f_n(t) \end{array}\right)=\left(\begin{array}{cccc} a_{11}(t) & a_{12}(t) & \cdots & a_{1n}{t}\\ a_{21}(t) & a_{22}(t) & \cdots & a_{2n}(t)\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1}(t) & a_{n2}(t) & \cdots & a_{nn}(t) \end{array}\right)\left(\begin{array}{c} g_1(t)\\ g_2(t)\\ \vdots\\ g_n(t) \end{array}\right)$$
where, let's say that the coefficients $a_{ij}(t)$ are allowed to have some dependence on $t$ (some of which might be constant anyway) but are 'nice' functions, e.g. simple exponential functions, nothing with any discontinuities or anything, is it still valid to find eigenvectors and eigenvalues by the same methods as for constant terms $a_{ij}$?
Yes. Your eigenthings will now themselves be functions of $t$, but if you're ok with that then have at it!