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We study the finite-step realizability of the joint/generalized spectralradius of a pair of real $d\times d$ matrices, one of which has rank 1. Then weprove that there always exists a finite-length word for which there holds thespectral finiteness property for the set of matrices under consideration. Thisimplies that stability is algorithmically decidable in our case.

The finite-step realizability of the joint spectral radius of a pair of $d\times d$ matrices one of which being rank-one