Non-commutative Sylvester's determinantal identity

Sylvester's identity is a classical determinantal identity with a straightforward linear algebra proof. We present a new, combinatorial proof of the identity, prove several non-commutative versions, and find a $\beta$-extension that is both a generalization of Sylvester's identity and the $\beta$-extension of the MacMahon master theorem.Comment: 28 pages, 8 figure

Combinatorics of determinantal identities

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 125-129).In this thesis, we apply combinatorial means for proving and generalizing classical determinantal identities. In Chapter 1, we present some historical background and discuss the algebraic framework we employ throughout the thesis. In Chapter 2, we construct a fundamental bijection between certain monomials that proves crucial for most of the results that follow. Chapter 3 studies the first, and possibly the best-known, determinantal identity, the matrix inverse formula, both in the commutative case and in some non-commutative settings (Cartier-Foata variables, right-quantum variables, and their weighted generalizations). We give linear-algebraic and (new) bijective proofs; the latter also give an extension of the Jacobi ratio theorem. Chapter 4 is dedicated to the celebrated MacMahon master theorem. We present numerous generalizations and applications. In Chapter 5, we study another important result, Sylvester's determinantal identity. We not only generalize it to non-commutative cases, we also find a surprising extension that also generalizes the master theorem. Chapter 6 has a slightly different, representation theory flavor; it involves representations of the symmetric group, and also Hecke algebras and their characters. We extend a result on immanants due to Goulden and Jackson to a quantum setting, and reprove certain combinatorial interpretations of the characters of Hecke algebras due to Ram and Remmel.by Matjaž Konvalinka.Ph.D

Numerical issues and computational problems in algebraic control theory

The work of this thesis concerns computational issues arising from various fields of Algebraic Control Theory. Efficient algorithms covering the following classes of problems are developed. (i) Exterior Algebra Computations: For given matrices [Please see formulas inside thesis] algorithms achieving the computation of [Please see formulas inside thesis] are formulated. An algorithm for the evaluation of Plucker matrices is also proposed. Most of these algorithms are used in the development of a unifying numerical algorithm for the solution of the Determinantal Assignment Problem. (ii) Numerical Techniques for handling nonqeneric computations: Several numerical tools for the diagnosis of certain properties in an "almost sense", and the definition of procedures attaining the termination of algorithms are developed. (iii) Evaluation of the Greatest Common Divisor of polynomials: A new numerical algorithm for the evaluation of the greatest common divisor of any set of polynomials is formulated. (iv) Almost Zero Computations: Algorithms achieving the evaluation of the Prime almost zero of a polynomial set and the computation of the zero radius are given. Useful comments about the achievement of improved bounds for the zero-trapping region are also presented

Soliton solutions of noncommutative integrable systems

This thesis is concerned with solutions of noncommutative integrable systems where the noncommutativity arises through the dependent variables in either the hierarchy or Lax pair generating the equation. Both Chapters 1 and 2 are entirely made up of background material and contain no new material. Furthermore, these chapters are concerned with commutative equations. Chapter 1 outlines some of the basic concepts of integrable systems including historical attempts at finding solutions of the KdV equation, the Lax method and Hirota's direct method for finding multi-soliton solutions of an integrable system. Chapter 2 extends the ideas in Chapter 1 from equations of one spatial dimension to equations of two spatial dimensions, namely the KP and mKP equations. Chapter 2 also covers the concepts of hierarchies and Darboux transformations. The Darboux transformations are iterated to give multi-soliton solutions of the KP and mKP equations. Furthermore, this chapter shows that multi-soliton solutions can be expressed as two types of determinant: the Wronskian and the Grammian. These determinantal solutions are then verified directly. In Chapter 3, the ideas detailed in the preceding chapters are extended to the noncommutative setting. We begin by outlining some known material on quasideterminants, a noncommutative KP hierarchy containing a noncommutative KP equation, and also two families of solutions. The two families of solutions are obtained from Darboux transformations and can be expressed as quasideterminants. One family of solutions is termed ``quasiwronskian'' and the other ``quasigrammian'' as both reduce to Wronskian and Grammian determinants when their entries commute. Both families of solutions are then verified directly. The remainder of Chapter 3 is original material, based on joint work with Claire Gilson and Jon Nimmo. Building on some known results, the solutions obtained from the Darboux transformations are specified as matrices. These solutions have interesting interaction properties not found in the commutative setting. We therefore show various plots of the solutions illustrating these properties. In Chapter 4, we repeat all of the work of Chapter 3 for a noncommutative mKP equation. The material in this chapter is again based on joint work with Claire Gilson and Jon Nimmo and is mainly original. The original material in Chapters 3 and 4 appears in \cite{gilson:nimmo:sooman2008} and in \cite{gilson:nimmo:sooman2009}. Chapter 5 builds on the work of Chapters 3 and 4 and is concerned with exponentially localised structures called dromions, which are obtained by taking the determinant of the matrix solutions of the noncommutative KP and mKP equations. For both equations, we look at a three-dromion structure from which we then perform a detailed asymptotic analysis. This aymptotic forms show interesting interaction properties which are demonstrated by various plots. This chapter is entirely the author's own work. Chapter 6 presents a summary and conclusions of the thesis