Froilan Dopico, "Polynomial and rational matrices with prescribed data"

Thursday 23 Jan 2025, 14:30 → 15:30 Europe/Rome

Ex-ISEF/Building-Main Lecture Hall (GSSI)

Abstract:

We study necessary and sufficient conditions for the existence of polynomial and rational matrices with different prescribed data. First, we consider the problem for polynomial matrices when the size, degree, rank, invariant factors, infinite elementary divisors, and the minimal indices of their left and right null spaces are prescribed and prove that a polynomial matrix with these data exists if and only if these data satisfy a surprisingly simple unique condition related to a fundamental constraint known as the "index sum theorem'". In the second place, we extend this result to rational matrices when the size, rank, invariant rational functions, invariant orders at infinity, and minimal indices of their left and right null spaces are prescribed. The data prescribed so far are called in the literature the "complete structural data" or the "complete eigenstructure" of the polynomial or rational matrix. Finally, in addition to the complete eigenstructure, we also prescribe the minimal indices of the row and column spaces and show that the simple condition found in the previous problems must be completed with a majorization relation among the involved indices, the degrees of the invariant factors and of the infinite elementary divisors.