The setting is the following. We are given a measure $\nu$ supported on the set of $d \times d$ matrices. We assume that $\nu$ is strongly irreducible, is proximal and has positive eventual rank.

We consider the partial products of a sequence of i.i.d matrices i.e, the sequence $(\overline\gamma_n)_n := (\gamma_0\cdots\gamma_{n-1})_n$ for $(\gamma_n)_n$ an i.i.d sequence of law $\nu$.

We will give an idea of the proof of a few nice results like the linear escape of some contraction coefficients up to an exponentially small probability, an exponentional mixing property for the action on the projective space as well as the convergence of the $n$-th root of coefficients with an additional moment assumption (which is optimal).

We will first quickly introduce the notions of strong irreducibility, proximality and eventual rank and write down statements of the results. Then we will describe the pivoting technique for random walks on a free group and give a geometric intuition on why this technique also works for products of matrices.