Minicurso: Perturbations of matrix pencils and polynomials
Ministrado por: Andrii Dmytryshyn
Data: 24, 29 e 31 de outubro
Horário: 10 às 12
Local: Sala Eliana Aude (localizada nas dependências do NCE)
Resumo: Complete eigenstructures of matrices, matrix polynomials and pencils are reflected in the corresponding canonical forms, e.g., Jordan and Kronecker canonical forms. These canonical forms are well known and studied with various purposes but the reductions to these forms are unstable operations: both the corresponding canonical forms and the reduction transformations depend discontinuously on the entries of an original matrix or matrix pencil. Therefore, V.I. Arnold introduced a normal form, with the minimal number of independent parameters, to which an arbitrary family of matrices A' close to a given matrix A can be reduced by similarity transformations smoothly depending on the entries of A'. He called such a normal form a miniversal deformation of A. Now the notion of miniversal deformations has been extended to various matrices and matrix pencils. Miniversal deformations can help us to construct stratifications, i.e., closure hierarchies, of orbits and bundles. These stratifications are the graphs that show which complete eigenstructures the matrices, matrix pencils or polynomials may have in an arbitrarily small neighbourhood of a given matrix, matrix pencil or polynomial. In particular, the stratifications show how the eigenvalues may coalesce or split apart, appear or disappear. The course will start with the classical results by V.I. Arnold on the perturbations of matrices and cover their recent extensions to the structured matrix systems. The second part of the course will be dedicated to the geometrical aspects of the perturbation theory.
About Andrii Dmytryshyn: Andrii research interests are in the fields of matrix analysis and computational mathematics. Before starting as an Associate Senior Lecturer at Örebro University, Andrii worked and studied in Umeå (Sweden), Bordeaux (France), Padua (Italy), and Kyiv (Ukraine).
Andrii received the SIAM Student Paper Prize 2015 (which is one of the major prizes awarded by the Society for Industrial and Applied Mathematics). He was selected as one of the top 7 candidates for the Householder Prize XX (The Householder Prize is an award for the best dissertation in numerical linear algebra in a 3-year period). Andrii was also an LAA Early Career Speaker at ILAS 2017 and invited and plenary speaker at several other conferences.
Andrii teaches courses on numerical methods for differential equations, applied mathematics, mathematical control theory, matrix computations, and case studies in computational mathematics.