  We introduce the notions of $\varepsilon$-approximate fixed point and weak
$\varepsilon$-approximate fixed point. We show that for a group of unitary
matrices even the existence of a nontrivial weak $\varepsilon$-approximate
fixed point for sufficiently small $\varepsilon$ gives an actual nontrivial
common eigenvector. We give estimates for $\varepsilon$ 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.
