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 $G_2$ and $G_3$ for the closure ordering on the sets of congruence classes of $2\times 2$ and $3\times 3$ complex matrices. In other words, we construct two directed graphs whose vertices are $2\times 2$ or, respectively, $3\times 3$ canonical matrices under congruence and there is a directed path from $A$ to $B$ if and only if $A$ can be transformed by an arbitrarily small perturbation to a matrix that is congruent to $B$. A bundle of matrices under congruence is defined as a set of square matrices $A$ for which the pencils $A+\lambda A^T$ belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of $2\times 2$ or $3\times 3$ matrices have the same properties with respect to perturbations. We construct the Hasse diagrams $G_2^{\rm B}$ and $G_3^{\rm B}$ for the closure ordering on the sets of congruence bundles of $2\times 2$ and, respectively, $3\times 3$ matrices. We find the isometry groups of $2\times 2$ and $3\times 3$ congruence canonical matrices.Comment: 34 page

Miniversal deformations of pairs of symmetric matrices under congruence

For each pair of complex symmetric matrices $(A,B)$ we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation that smoothly depends on the entries of $\widetilde{A}$ and $\widetilde{B}$. Such a normal form is called a miniversal deformation of $(A,B)$ 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 $(A,B)$ 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