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It is widely believed that typical finite families of $d \times d$ matricesadmit finite products that attain the joint spectral radius. This conjecture issupported by computational experiments and it naturally leads to the followingquestion: are these spectrum maximizing products typically unique, up to cyclicpermutations and powers? We answer this question negatively. As discovered byHorowitz around fifty years ago, there are products of matrices that alwayshave the same spectral radius despite not being cyclic permutations of oneanother. We show that the simplest Horowitz products can be spectrum maximizingin a robust way; more precisely, we exhibit a small but nonempty open subset ofpairs of $2 \times 2$ matrices $(A,B)$ for which the products $A^2 B A B^2$ and$B^2 A B A^2$ are both spectrum maximizing.

Spectrum maximizing products are not generically unique