Minimal bases of matrix pencils: Algebraic toeplitz structure and geometric properties

Abstract

For a singular matrix pencil sF− G the structure of the right rational null space H ̃ r is studied and new algebraic and geometric properties and invariants are established. The results highlight the nature of the isomorphism between polynomial vectors of H ̃ r and real vectors in the right spaces { N k r, k ⩾0} , defined as null spaces of Toeplitz matrices of ( F, G). The algebraic structure of L ̃ r is described by the properties of ordered minimal bases, as well as those of new invariants, the prime R[ s]-modules { M i, i ∈ μ ̃ } The geometric structure of H ̃ r is defined by the properties of new invariant spaces, P i , P ̂ i , R i , associated with M i,i μ ̃ m—the high, low, and prime spaces—as well as some additional invariant spaces defined through the Toeplitz representation of ordered minimal bases. These new invariant spaces and modules are shown to be naturally defined also on the right spaces N k r , and this provides an alternative geometric definition of them, which is independent of the original algebraic one. It is shown that the construction of an ordered minimal basis is equivalent to a standard linear algebra problem, the selection of a right system of generators for { N k r, k ⩾ 0} . Finally, certain parametrization issues for the polynomial vectors of H ̃ r are considered.