The finite-step realizability of the joint spectral radius of a pair of   $d\times d$ matrices one of which being rank-one
  We study the finite-step realizability of the joint/generalized spectral radius of a pair of real $d\times d$ matrices, one of which has rank 1. Then we prove that there always exists a finite-length word for which there holds the spectral finiteness property for the set of matrices under consideration. This implies that stability is algorithmically decidable in our case. 
Xiongping Dai
math.OC cs.SY math.DS