2017/12/07 by Dopico, Froilán M., Van Dooren, Paul
#15A54 #15A60 #15B05 #65F15 #65F35 #93B18 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1712.03816
This paper studies generic and perturbation properties inside the linear space of m× (m+n) polynomial matrices whose rows have degrees bounded by a given list d1, …, dm of natural numbers, which in the particular case d1 = ⋯ = dm = d is just the set of m× (m+n) polynomial matrices with degree at most d. Thus, the results in this paper extend to a much more general setting the results recently obtained in [Van Dooren & Dopico, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.011] only for polynomial matrices with degree at most d. Surprisingly, most of the properties proved in [Van Dooren & Dopico, Linear Algebra Appl. (2017)], as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to d1, … , dm, and with right minimal indices differing at most by one and having a sum equal to ∑i=1m di, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly.