Computing the change of the length of this vector

Question

I have a problem of computing the length of varying vector.

Consider an arbitary 2-dimensional vector $y_0$. Start at $y_0$, define $y_{t+1}$ by: $y_{t+1} = Ay_t$,

Here $A$ denotes the matrix:

$$ \begin{pmatrix} 0.8 & -0.2 \\ 1.6 &0.3 \\ \end{pmatrix} $$

My question is, as $t\to \infty$, what does the length $lim_{t\to\infty}|y_{t}|$ goes to? Would it go towards infinity or zero?

I think the answer may depends on $y_0$, to be more precise, depends on whether we have a component of $y_0$ is zero. And I guess there would be a way of explain the process of changing in the sense of the meaning of determinant. Could someone please pointing out a way of thinking about this? May I please ask for a solution or some reference?

Thanks in advance!

Answer 1

One way to proceed is to diagonalize the matrix. I assume you are familiar with that.

Once you have done this, note that $$A = VDV^{-1}\Rightarrow A^k = VD^kV^{-1}$$ for all $k\in \mathbb N$. And because $D$ is diagonal, powers are entrywise.

With this in hand you can compute the limit:

$$\lim_{n\to\infty}A^n y_0$$