Products of $2 \times 2$ Random Matrices

Abstract

The notion of the shape of a triangle can be used to define the shape of a $2 \times 2$ real matrix; we find that the shape of a matrix retains just the right amount of information required for determining the main features of the asymptotic behaviour, as $n\rightarrow\infty$, of $\mathbf{G}_n\mathbf{G}_{n-1}\cdots\mathbf{G}_1$, where the $\mathbf{G}_i$ are i.i.d. copies of a $2 \times 2$ random matrix $\mathbf{G}$. An alternative formula to the Furstenberg formula is proposed for the upper Lyapounov exponent of the probability distribution of $\mathbf{G}$. We find that in some cases, using our formula, the Lyapounov exponent is more susceptible to explicit calculation.