Hermite indices and equivalence relations. (Q1426300)

The Hermite indices, defined by considering linear dependence patterns in the columns of the controllability matrix of a linear system, form a system of invariants under similarity transformations. In the frequency-domain setting, Hermite indices correspond to the degrees of the diagonal entries of a polynomial matrix reduced to the canonical Hermite form. Here, the authors study equivalence relations other than similarity transformations, and show that in general the Hermite indices are not invariant. Then, they carefully describe a class of equivalence relations for which the Hermite indices remain invariant. The study is carried out both in the state-space and frequency, or polynomial domains.