On Strongly Non-singular Polynomial Matrices

Abstract

We consider matrices with infinite power series as entries and suppose that those matrices are represented in an “approximate” form, namely, in a truncated form. Thus, it is supposed that a polynomial matrix P which is the l-truncation (l is a non-negative integer, $\deg P=l$) of a power series matrix M is given, and P is non-singular, i.e., $\mathrm{det\,}P\ne 0$. We prove that the strong non-singularity testing, i.e., the testing whether P is not a truncation of a singular matrix having power series entries, is algorithmically decidable. Supposing that a non-singular power series matrix M (which is not known to us) is represented by a strongly non-singular polynomial matrix P, we give a tight lower bound for the number of initial terms of $M^{-1}$ which can be determined from $P^{-1}$. In addition, we report on possibility of applying the proposed approach to “approximate” linear differential systems.