Some Families of Combinatorial Matrices and Their Algebraic Properties

Abstract

In this thesis, we will study some properties of certain families of combinatorial matrices. While some of the families will be examined throughout this thesis are new and firstly investigated, the others are the generalizations of some of the previously known matrices. We gather our studies into six different groups. They are non-symmetric band matrices with Gaussian q-binomial entries, generalization of the super Catalan matrix, families of Max and Min matrices, a non-symmetric variant of the Filbert matrix, a nonlinear generalization of the Filbert matrix and some certain Hessenberg matrices. For all matrices will be studied except the Hessenberg matrices, we present explicit formulae for the LU-decompositions, determinants, inverse and LU-decompositions of the inverses of the matrices as well as the Cholesky decompositions when the matrix is symmetric. Additionally, we evaluate some certain Hessenberg determinants via generating function method. We use some new and existing methods to prove our claims. Particularly, we present a new method to evaluate determinants of some Hessenberg matrices whose entries consist of terms of higher order linear recursive sequences.