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.