# O redukcijama sistema linearnih operatorskih jednačina

Title: | O redukcijama sistema linearnih operatorskih jednačina |

Author: | Jovović, Ivana |

Abstract: | 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 + '1 A(x2) = b21x1 + b22x2 + : : : + b2nxn + '2 ... A(xn) = bn1x1 + bn2x2 + : : : + bnnxn + 'n; where B = [bij ]n n is matrix over the eld K, A is linear operator on the vector space V over K and where '1; '2; : : : ; 'n 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 B( ) of the matrix B and recurrence for the coe cients of the adjugate matrix of the characteristic matrix 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 rst 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 nite dimensional vector space V over K as module over the ring K[x] of polynomials in x with coe cients 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 rst 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 v in two or three unknowns and then we study reductions of systems in n unknowns. The partial reduction requests changing of basis so that the system matrix is in the rational or Jordan form. We also treat the total reduction of the obtained partially reduced systems in this manner. Subsection 4.4. "Total Reduction for Linear Systems of Operator Equations with System Matrix in Companion Form" is one interesting way to proceed consideration started in previously mentioned works. It is based on papers of L. Brand [3, 4]. The fth section is generalization of the forth. Here we examine systems in n unknowns and with di erent linear operators. We introduce the notion of characteristic polynomial in more than one unknown - generalized characteristic polynomial and a method for total reduction by nding adjugate matrix of the generalized characteristic matrix of the system matrix. The sixth section is a summary of applications and examples of methods for partial and total reduction. There are some examples of the rst and higher order linear systems of di erential equations and di erent approaches for calculating rational and Jordan canonical forms. The last section is devoted to the study of di erential transcendence of the solution of the rst order linear system of di erential equations with complex coe cients, where exactly one of the following meromorphic functions '1; '2; : : : ; 'n is di erentially transcendental, using method of total reduction. We review some of the standard facts on di erential transcendence following books [45, 44, 39, 50, 7, 33, 15, 23, 36]. |

URI: | http://hdl.handle.net/123456789/2585 |

Date: | 2013 |

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