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I am working with some simple systems of difference equations. For example, writing it in matrix form it's:

\(\displaystyle \left ( \begin{matrix} a_{n + 1} \\ b_{n + 1} \end{matrix} \right ) = \left ( \begin{matrix} 0 & n \\ 1 & 0 \end{matrix} \right ) \left ( \begin{matrix} a_n \\ b_n \end{matrix} \right )\)

The idea is to use matrix methods to simplify the system, ie if I diagonalize the matrix and transform the vectors into the eigenbasis I can separate the a's and b's. So I do that, solve the matrix recursion and transform back to the original basis. Done, right? Wrong! The solution doesn't check.

I know we can do this with a system with constant matrix elements so my question is: Does the fact that we have a non-constant transformation matrix (ie it depends on n) change the recursion when we change to the eigenbases?

-Dan