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

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