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Abstract:
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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]. |