# Finding the limits to sequences given by recursive transformation of a vector: $a_n = a_{n-1}M$

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?

• This can be answered by studying the eigenvalues of $M$. If $M$ has a complete set of eigenvectors $e_i$ with eigenvalues $\lambda_i$, which is the case if $M$ is symmetric for example, then the initial vector can be expanded as $a_0 = \sum c_n e_n$ which gives $a_n = M^n a_{0} = \sum c_k M^ne_k = \sum c_k \lambda_k^n e_k$. As $n\to\infty$ this sum will be dominated by the terms with largest eigenvalue. If this eigenvalue is negative then $a_n/\|a_n\|$ diverges otherwise it converges. The analysis is a bit more involved if the eigenvalues are degenerate. – Winther Jun 6 '16 at 18:11
• When the matrix $M$ is stochastic then the sequence $a_n = Ma_{n-1}$ is known as a Markov chain. It has a rich litterature, just try searching for it. – Winther Jun 6 '16 at 18:20