Rank one matrices do not contribute to the failure of the finiteness property

The joint spectral radius of a bounded set of d times d real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. A set of matrices is said to satisfy the finiteness property if this maximum rate of growth occurs along a periodic infinite sequence. In this note we give some sufficient conditions for a finite set of matrices to satisfy the finiteness property in terms of its rank one elements. We show in particular that if a finite set of matrices does not satisfy the finiteness property, then the subset consisting of all matrices of rank at least two is nonempty, does not satisfy the finiteness property, and has the same joint spectral radius as the original set. We also obtain an exact formula for the joint spectral radii of sets of matrices which contain at most one element not of rank one, generalising a recent result of X. Dai.