Doctor of Philosophy (Ph.D.)
In this dissertation, we study three different sets of matrices. First, we consider Euclidean distance squared matrices. Given n points in Euclidean space, we construct an n x n Euclidean squared distance matrix by assigning to each entry the square of the pairwise interpoint Euclidean distance. The study of distance matrices is useful in computational chemistry and structural molecular biology. The purpose of the first part of the thesis is to better understand this set of matrices and its different characterizations so that a number of open problems might be answered and known results improved. We look at geometrical properties of this set, investigate forms of linear maps that preserve this set, consider the uniqueness of completions to this set and look at subsets that form regular figures. In the second part of this thesis, we consider ray-pattern matrices. A ray-pattern matrix is a complex matrix with each nonzero entry having modulus one. A ray-pattern is said to be ray-nonsingular if all positive entry-wise scalings are nonsingular. A full ray-pattern matrix has no zero entries. It is known that for n > 5, there are no full ray-nonsingular matrices but examples exist for n < 5. We show that there are no 5 x 5 full ray-nonsingular matrices. The last part of this thesis studies certain of the finite reflection groups. A reflection is a linear endomorphism T on the Euclidean space V such that T(v) = v - 2(v, u)u for all v ∈ V. A reflection group is a group of invertible operators in the algebra of linear endomorphism on V that are generated by a set of reflections. One question that has recently been studied is the form of linear operators that preserve finite reflection groups. We first discuss known results about preservers of some finite reflection groups. We end by showing the forms of the remaining open cases.
© The Author
Milligan, Thomas W., "On certain sets of matrices: Euclidean squared distance matrices, ray-nonsingular matrices and matrices generated by reflections" (2004). Dissertations, Theses, and Masters Projects. Paper 1539623440.