  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.
