On pairs of spectrum maximizing products with distinct factor multiplicities

Recently, Bochi and Laskawiec constructed an example of a set of matrices $\{A,B\}$ having two different (up to cyclic permutations of factors) spectrum maximizing products, $AABABB$ and $BBABAA$. In this paper, we identify a class of matrix sets for which the existence of at least one spectrum maximizing product with an odd number of factors automatically entails the existence of another spectrum maximizing product. Moreover, in addition to Bochi--Laskawiec's example, the number of factors of the same name (factors of the form $A$ or $B$) in these matrix products turns out to be different. The efficiency of the proposed approach is confirmed by constructing an example of a set of $2\times2$ matrices $\{A,B\}$ that has spectrum maximizing products of the form $BAA$ and $BBA$.