The finite-step realizability of the joint spectral radius of a pair of d×dd×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 square matrices S1S1 and S2S2, one of which has rank 1, where 2⩽d<+∞2⩽d<+∞. Let ρ(A)ρ(A) denote the spectral radius of a square matrix A  . Then we prove that there always exists a finite-length word (i1∗,…,im∗)∈{1,2}m, for some finite m⩾1m⩾1, such thatρSi1∗⋯Sim∗m=supn⩾1max(i1,…,in)∈{1,2}nρ(Si1⋯Sin)n.In other words, there holds the spectral finiteness property for {S1,S2}{S1,S2}. Explicit formula for computation of the joint spectral radius is derived. This implies that the stability of the switched system induced by {S1,S2}{S1,S2} is algorithmically decidable in this case.