Affordable Access

Publisher Website

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

Linear Algebra and its Applications
Publication Date
DOI: 10.1016/0024-3795(94)90371-9
  • Mathematics


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.

There are no comments yet on this publication. Be the first to share your thoughts.


Seen <100 times

More articles like this

The algebraic structure of pencils and block Toepl...

on Linear Algebra and its Applica... Jan 01, 1998

Minimal singularities in orbit closures of matrix...

on Linear Algebra and its Applica... Jan 01, 2003

Algebraic properties of Toeplitz operators on the...

on Journal of Mathematical Analys... Jan 01, 2007

Grassmann invariants, matrix pencils, and linear s...

on Linear Algebra and its Applica... Jan 01, 1996
More articles like this..