writing a recursion relation to a matrix

Question

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.

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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

Answer 1

You could write $$ \begin{pmatrix} X_{t+1} \\ V_{t+1} \\ 1\end{pmatrix} = \begin{pmatrix} 1-cy & w & cyg \\ -cy & w & cyg \\ 0 & 0 & 1\end{pmatrix} \begin{pmatrix}X_t \\ V_t \\ 1\end{pmatrix} $$ This way you essentially have three sequences $X_t$, $V_t$ and $Z_t$, with $Z_t=1$ for all $t$. Now you can analyze their behaviour using this new recursive relation and the knowledge that $Z_t=1$.

Answer 2

Let $Y_t=[X_t,V_t]^T,A=\begin{pmatrix}1-cy&w\\-cy&w\end{pmatrix}$. Then $Y_{t+1}=AY_t+cyg[1,1]^T$. If $cyg=0$, then $Y_t=A^tY_0$.

Assume that $cyg\not=0$. Since $I-A$ is invertible, there is a unique fixed point $T$ s.t. $T=AT+cyg[1,1]^T$. Put $Z_t=Y_t-T$; then $Z_{t+1}=AZ_t$ and $Z_t=A^tZ_0$.

Answer 3

As slight variation, write $X_{t+1}$ and $V_{t+1}$ in terms of $X_t$ and $V_t$:

$$ X_{t+1} = X_t + w V_t - cy X_t + cyg\\ V_{t+1} = wV_t -cyX_t + cgy $$ notice that the term $cgy$ is independent of $t$, the relation is not a matrix relation, it is an affine recursive relation which can be written as:

$$ \left(\begin{array}{c} X_{t+1} \\ V_{t+1} \end{array}\right) = \left( \begin{array}{cc} (1-cy) & w \\ -cy & w \end{array} \right)\left(\begin{array}{c} X_t \\ V_t \end{array}\right) + \left(\begin{array}{c} cyg \\ cyg \end{array}\right) $$ This is a relation of the type $Q_{t+1} = A Q_{t} + Y$ and can be solved by the change of variable to $Z_t= Q_t + \Lambda$ where the matrix $\Lambda$ is chosen to make the recursive relation a matrix relation (instead of affine), this is achieved by taking $\Lambda = - ({\bf 1} -A)^{-1} Y$, one gets:

$$ Z_{t+1} = A Z_{t} = A^{t}Z_1 $$