44 research outputs found
Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence
We construct the Hasse diagrams and for the closure ordering on
the sets of congruence classes of and complex matrices.
In other words, we construct two directed graphs whose vertices are
or, respectively, canonical matrices under congruence and there is
a directed path from to if and only if can be transformed by an
arbitrarily small perturbation to a matrix that is congruent to .
A bundle of matrices under congruence is defined as a set of square matrices
for which the pencils belong to the same bundle under
strict equivalence. In support of this definition, we show that all matrices in
a congruence bundle of or matrices have the same
properties with respect to perturbations. We construct the Hasse diagrams
and for the closure ordering on the sets of
congruence bundles of and, respectively, matrices. We
find the isometry groups of and congruence canonical
matrices.Comment: 34 page
Miniversal deformations of pairs of symmetric matrices under congruence
For each pair of complex symmetric matrices we provide a normal form
with a minimal number of independent parameters, to which all pairs of complex
symmetric matrices , close to can be
reduced by congruence transformation that smoothly depends on the entries of
and . Such a normal form is called a miniversal
deformation of under congruence. A number of independent parameters in
the miniversal deformation of a symmetric matrix pencil is equal to the
codimension of the congruence orbit of this symmetric matrix pencil and is
computed too. We also provide an upper bound on the distance from to
its miniversal deformation.Comment: arXiv admin note: text overlap with arXiv:1104.249
A Geometric Description of the Sets of Palindromic and Alternating Matrix Pencils with Bounded Rank
The sets of n x n T-palindromic, T-antipalindromic, T-even, and T-odd matrix pencils with rank at most r < n are algebraic subsets of the set of n x n matrix pencils. In this paper, we determine their dimension and we prove that they are all irreducible. This is in contrast with the nonstructured case, since it is known that the set of n x matrix pencils with rank at most r< n is an algebraic set with r + 1 irreducible components. We also show that these sets of structured pencils with bounded rank are the closure of the congruence orbit of a certain structured pencil given in canonical form. This allows us to determine the generic canonical form of a structured n x n matrix pencil with rank at most r, for any of the previous structures
Generic symmetric matrix pencils with bounded rank
We show that the set of n × n complex symmetric matrix pencils of rank at most r is the union of the closures of [r/2] + 1 sets of matrix pencils with some, explicitly described,complete eigenstructures. As a consequence, these are the generic complete eigenstructures of n × n complex symmetric matrix pencils of rank at most r. We also show that these closures correspondto the irreducible components of the set of n × n symmetric matrix pencils with rank at most r when considered as an algebraic set