I have a difference equation in the form Un = AUn-1 and I am wondering how does the stability of the system depend on the eigenvalues of the matrix A?
My question relates to modelling pig (bonhams) growth over time given a set of initial conditions. My sequence of values for the different seasons in 1 1 7 13 55 and so on and so on.
My A matrix looks as follows [1 6 / 1 0] (excuse the poor formatting) therefore my eigenvalues are 3 and -2.
Can anyone please intuitively explain how these values impact the stability of the population.
From my research so far I've ready about how one of the eigenvalues controls the oscillations around steady state but I don't fully get it.
Also how can I find the stable age distribution of this matrix, is it a case of normalising the eigenvector for one of the eigenvalues to 1?
The solution to the problem
$$ u_n = A u_{n - 1} $$
is
$$ u_n = A^n u_0 = (V \Lambda V^{-1})^n u_0 = V\Lambda^nV^{-1}u_0 $$
where $V$ is the matrix containing the eigenvectors of $A$ and $\Lambda$ is a diagonal matrix with the eigenvalues of $A$
$$ \Lambda = {\rm diag}(\lambda_1, \lambda_2, \cdots) $$
Since
$$ \Lambda^n = {\rm diag}(\lambda_1^n,\lambda_2^n,\cdots) $$
The solution $u_n$ will blow up if there is a $|\lambda_k| > 1$