EXPONENTIAL FUNCTION ON MATRIX GROUPS

V diplomskem delu obravnavamo eksponentno funkcijo na matričnih grupah. Prvo poglavje je namenjeno vpeljavi grup iz matrik s kompleksnimi koeficienti. Te grupe, ki so hkrati tudi Liejeve grupe, in poznavanje njihovih lastnosti je temeljnega pomena pri vpeljavi osrednjih matematičnih struktur v diplomskem delu. Sledi konstrukcija bilinearnih form, tako na poljubnih vektorskih prostorih, kot na na matričnih grupah. Glede na njih opredelimo simplektične ter ortogonalne matrične grupe. V drugem poglavju ilustriramo geometrično strukturo na matrikah z vpeljavo pojmov metrika in norma. V nadaljevanju definiramo eksponentno funkcijo na matrikah. Seznanimo se z dvema metodama, s pomočjo katerih lahko izračunamo vrednost matrične funkcije: z metodo razvoja v potenčno vrsto in metodo diagonalizacije dane matrike. Obe metodi podkrepimo s primeri.This diploma thesis discusses exponential function on matrix groups. The first part introduces matrix groups with complex coefficients. These groups that are also Lie groups and knowledge of their characteristics are of the greatest significance by introduction of central mathematical structures in this thesis. Then a construction of bilinear forms follows, both in arbitrary vector space and in matrix groups. With regard to these we define orthogonal and symplectic groups. The second part illustrates geometric structure on matrices with introduction of the concepts of metrics and norm. In the following we define exponential function on matrix groups. We introduce two methods, by means of which we calculate the value of matrix function: by method of development in power series and method of diagonalization of given matrix. Both methods are illustrated by concrete examples