On approximate and actual reducibility of matrix groups

We introduce the notions of $\epsilon$-approximate fixed point and weak $\epsilon$-approximate fixed point. We show that for a group of unitary matrices even the existence of a nontrivial weak $\epsilon$-approximate fixed point for sufficiently small $\epsilon$ gives an actual nontrivial common eigenvector. We give estimates for $\epsilon$ in terms of the size $n$ of matrices and prove that the dependence is polynomial. Moreover, we show that the common eigenvector is polynomially close to the starting weak approximate fixed point.