# How to operate with linear system on matrix Fourier coefficients — where to read about, if it's too simple?

Context: let's introduce a function to find: $$Z: \mathbb{C} \to \mathbb{C}^{3\times 3}$$. It satisfies linear DE: $$\dot{Z}(t) = A(t) Z(t)$$. Physical sense of problem says that $$Z$$ is periodic. Matrix $$A(t)$$ includes $$e^{\alpha t}$$, so to describe the solution, I decided to expand $$Z$$ to Fourier series with coefficients $$Z_k$$, $$k=..., -1, 0, 1, ...$$ That turned my DE into a chain of linear systems on $$Z_k$$ of the following form: $$Z_k = B Z_k + C Z_{k-1} + D Z_{k+1} + Z_k E$$.

My problem: If considering all elements of all $$Z_k$$ vectorised in one infinite vector, I have ordinary linear system with like 7-diagonal matrix. I can simplify it to get matrix with repeating $$3\times 3$$ blocks in diagonal with only change that true diagonal has terms proportional to $$k$$, but each $$k$$-th block will have one non-zero coefficient in the right touching block (to connect $$Z_k$$ with $$Z_{k+1}$$) and one in the left, similarly. This does not give a hope to split system into finite parts, but I want at least split this system into two systems, each determining even/odd part of function $$Z$$ to study the solution instead of this huge system. And I'm stuck here with a technical stupor: how to do it?

Excuse: This might be something that every student knows, but 1) I'm young student and may missed the source to study how to do it from, 2) I've got interested in DEs and faced with this issue in course of research, so need it a lot. So please if you consider it to be trivial, recommend where to study and train this kind of things.

• I presume your coefficient $\alpha$ is imaginary? (otherwise the solution will not be periodic in $t$) Oct 23, 2023 at 19:56
• Maybe better to move this to Math Stack Exchange? Oct 23, 2023 at 21:29