Unit:
Κατεύθυνση Εφαρμοσμένα ΜαθηματικάLibrary of the School of Science
Author:
Gryparis Panteleimon
Supervisors info:
Μαριλένα Μητρούλη, Αναπληρώτρια Καθηγήτρια, Τμημ. Μαθηματικών ΕΚΠΑ
Original Title:
Πίνακες Bézout και εφαρμογές
Translated title:
Bézout matrices and their applications
Summary:
The matrices are widely used in many fields of mathematics, especially in Linear Algebra and Arithmetic Analysis. A well-known and simple application of the matrices is to solve a system of linear equations. If a matrix is square, it is possible to deduce some of the properties of calculating its determinant. If the matrix is symmetric, then we have additional important properties.
This thesis researches Bézout matrices and their applications. The important element of these matrices is that they are symmetric, which it gives us a great advantage in applications comparing to other matrices. In addition these matrices reduce our complexity.
The first part presents the mathematical tools, which are
useful for the calculation and applications of Bézout matrices.
In the second part we present the definition of Bézout matrices, theoretical and numerical examples and the properties. Finally, the calculation functions of these tables via two numerical computing environments, Matlab (version R2015a) and Maple (version 2016) are introduced.
The third part consists of several theorems which connect the Bézout matrices with the Greatest Common Divisor (GCD) of two univariate polynomials. It is possible to calculate both the degree and the coefficients of GCD via the application of these theorems. Furthermore many examples are provided to verify these theorems.
In the final two sections a number numerical applications are presented, the conclusions of this thesis, and the use of both these matrices and GCD of two univariate polynomials. It is worth mentioning that the goal is to calculate the GCD via Bézout matrix so that the complexity will be of the order O(n^2).
Main subject category:
Science
Other subject categories:
Μαθητικά Θέματα
Mathematics
Keywords:
matrices, Bézout, Bézout matrices, polynomials, univariate polynomials, Greatest Common Divisor of polynomials, GCD
