I'm interested in finding limits to 'recursive vector sequences' of the form $a_n = a_{n-1}M$ (where $a$ is a vector and the matrix $M$ is a transformation of $a$) but I don't know where to read about this topic or what it' called.
Since we are talking about vectors, it seems we actually have two limits to consider: $\lim\limits_{n\to\infty}(a_n/||a_n||)$, the limit of the 'direction' of $a_n$, and $\lim\limits_{n\to\infty}||a_n||$, the limit of the magnitude.
For example, $$M = \begin{bmatrix} 0.5 & 0 \\ 0 & 0.5 \end{bmatrix},\; a_0 = (1, 1) \longrightarrow \lim\limits_{n\to\infty}\frac{a_n}{||a_n||} = \frac1{\sqrt2}(1,1),\quad \lim\limits_{n\to\infty}||a_n||=0$$
And, in a second example, the direction of $a_n$ doesn't converge, $$M = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix},\; a_0 = (1, 0) \longrightarrow \lim\limits_{n\to\infty}||a_n||=1$$
So, how can we find the limits (and whether they exist) for non-trivial sequences of this form? Also, what notable properties can these sequences have (or what varieties do they come in), and where can I learn more?