The main purpose of this thesis is to define the stability of a system of linear difference equations of the form, ∇y(t) = Ay(t), and to analyze the stability theory for such a system using the eigenvalues of the corresponding matrix A in nabla discrete calculus and nabla fractional discrete calculus. Discrete exponential functions and the Putzer algorithms are studied to examine the stability theorem. This thesis consists of five chapters and is organized as follows. In the first chapter, the Gamma function and its properties are studied. Additionally, basic definitions, properties and some main theorem of discrete calculus are discussed by using particular example. In the second chapter, we focus on solving the linear difference equations by using the undetermined coefficient method and the variation of constants formula. Moreover, we establish the matrix exponential function which is the solution of the initial value problems (IVP) by the Putzer algorithm.

This dissertation presents the solution of difference equations and focuses on a method of difference equations solution with the aid of eigenvectors. The first part reminds the basic terms from area of difference equations such as dynamic of difference equations and linear difference equations of first order and higher order. Then the second section recalls also the system of difference equations including the fundamental matrix and general solution description. Afterthat, the method of solving the difference equations with a variation of constants and transform of scalar equations to the system are shown. The second part of the dissertation analyses some known algorithms and methods for the solution of linear difference equations. The Z-transform, its importance and usage for finding the solution of difference equation is recalled. Then the discrete analogue of Putzer's algorithm is mentioned because this algorithm was often used to check the results obtained by the newly described algorithm in further parts of this thesis. Also some ways of the system matrix power are stated. The next section then describes the principle of Weyr's method which is the basic point for further development of the theory including the presentation of the research results gained by Jiří Čermák in this area. The third part describes own solution of the difference equations system via eigenvectors based on the principle of Weyr's method for differential equations. The solution of system of linear homogeneous difference equtions with constant coefficients including the proof is presented and this solution is then extended to nonhomogeneous systems. Consequently to the theory, the influence of a nulity and the multiplicity of roots on the form of the solution is discussed. The last section of this part shows the implementation of the algorithm in Matlab program (for basic simpler cases) and its application to some cases of difference equations and systems with these equations. The final part of the thesis is more practical and it presents the usage of the designed algorithm and theory. Firstly, the algorithm is compared with Z-transform and the method of variation of constants and it is illustrated how to obtain the same results by using these three approaches. Then an example of current response solution in RLC circuit is demonstrated. The continuous case is solved and then the problem is transferred to discrete case and solved with the Z-transform and the method of eigenvectors. The obtained results are compared with the result of the continuous case.