On reduction formulas for linear systems of operator equations

This dissertation deals with an application of some linear algebra techniques for solving problems of reduction of system of linear operator equation of the form A(x1) = b11x1 + b12x2 +...+ b1nxn + φ1; A(x2) = b21x1 + b22x2 +...+ b2nxn + φ2; ... A(xn) = bn1x1 + bn2x2 +...+ bnnxn + φn; where B = [bij]nxn is matrix over the field K, A is linear operator on the vector space V over K and where φ1, φ2,..., φn are vectors in V . In particular, we consider reduction of such system under the action of the general linear group GL(n,K) and also reduction by using the characteristic polynomial ΔB(λ) of the matrix B and recurrence for the coefficients of the adjugate matrix of the characteristic matrix λ•I - B of the matrix B. The idea is to use rational and Jordan canonical forms to reduce the linear system of operator equations to an equivalent partially reduced system, i.e. to decompose the initial system into several uncoupled systems. This represents a new application of doubly companion matrix introduced by J.C. Butcher in [5]. In this work we are also concerned with transformation of the linear system of operator equation into totally reduced system, i. e. completely decoupled system of higher order linear operator equations. This results are related to results given by T. Downs in [13]. The thesis consists of two parts. The first part deals with properties of rational and Jordan canonical form. We start with Fundamental Theorem of Finitely Generated Modules Over a Principle Ideal Domain. If we consider finite dimensional vector space V over K as module over the ring K[x] of polynomials in x with coefficients in K, the Fundamental Theorem implies that there is a basis for V so that the associated matrix for B is in rational or Jordan form. The first section is adapted from Abstract Algebra of D. S. Dummit i R. M. Foote [14]. In the second section we look more closely at Hermite, Smith, rational and Jordan form and establish the relation between them. The structure of the similarity transformation matrix is also described. Some of theorem are considered from several aspects. This section provides a detailed exposition of normal forms using [14, 19, 37, 34, 35, 20, 57] and [1, 22, 48, 52, 54, 60]. The second part concerns with author's original contribution and it relies on papers [42, 43]. First we illustrate methods of the partial and the total reduction of systems in two or three unknowns and then we study reductions of systems in n unknowns. The partial reduc