Professor De Terran will talk about: 'New results on the inclusion of closure orbits and bundles of matrices and matrix pencils'
Orbits of nxn matrices under similarity are sets of matrices with the same Jordan Canonical form (JCF). When computing the JCF (or just the eigenvalues) of a matrix, the knowledge of all possible JCFs of small perturbations of a given JCF can help to understand the output of the algorithm, which is affected by roundoff errors.
The JCFs that can be obtained after small perturbations of a given JCF, say J, correspond to orbits that ``dominate" the orbit of J. In other words, the orbit of J is in the closure of its dominant orbits. The hierarchy of orbit closures of general matrices is well-known, as well as that of the set of matrices with bounded rank.
For matrix pencils (namely, pairs of matrices with the same size) the inclusion relationship between orbit closures has been also considered since, at least the 1980's. In this case, the standard equivalence relation is the so-called strict equivalence, which preserves the eigenstructure of the pencil, and the canonical form for this relation is the Kronecker canonical form (KCF). The hierarchy of orbit closures of general pencils under strict equivalence is also well-known. However, when the pencil has some particular structure (e. g., symmetric or Hermitian) then we encounter a different problem if we want the perturbations to maintain this structure. Some effort has been devoted in recent years to the analysis of orbit closures of structured pencils.
In this talk, we will review some recent results on the inclusion relationship between orbit closures of general and bounded-rank structured matrix pencils. We will also consider the inclusion relation of bundle closures. Bundles are generalizations of orbits, allowing the eigenvalues to change, while keeping the KCF.
