I have a recursion relation in the form of the following two equations:
$X_{t+1} = X_t + V_{t+1} \\ V_{t+1} = wV_t + cy(g-X_t)$
I want to write these two equations into a matrix form so that I can analyze using some Linear algebra techniques. the problem is, I cannot seem to write as a square matrix.
Here is my partial solution:
$X_{t+1} = X_t + wV_t + cy(g-X_t)\\ V_{t+1} = wV_t + cy(g-X_t)$
I will group all $X_t$ and $V_t$ together:
$X_{t+1} = (1-cy)X_t + wV_t + cyg \\ V_{t+1} = -cyX_t + wV_t + cyg$
If I set $g=0$, then this would come into a nice clean form as follows:
$X_{t+1} = (1-cy)X_t + wV_t \\ X_{t+1} = -cyX_t + wV_t$
So this can be written as follows:
$\begin{pmatrix} X_{t+1} \\ V_{t+1} \end{pmatrix} = \begin{pmatrix} 1-cy & w \\ -cy & w\end{pmatrix} \begin{pmatrix}X_t \\ V_t\end{pmatrix}$
The matrix therefore would be easy to diagonalize, etc. But we made a simplying case when $g=0$. Can I still make this form for the case when $g \neq 0$? How can I change the matrix?
Thanks