Matrix Completions for Commutativity
Authors:Jacob Buchholz, Wesley Chang, Maricela Cruz, Leah Jean-Louis, Marilyn Vasquez
Mentor:Geoffery Buhl, Associate Professor of Mathematics, California State University Channel Islands
Abstract: A partial matrix pattern is a set of entries in a matrix that are locations for specified entries and unspecified entries. Specified entries are considered to be arbitrary and fixed, yet unspecified entries are free to be chosen in order to satisfy some property for the matrix as a whole. In particular we consider the tricky case of completing a partial matrix pattern B with respect to a fully specified matrix A so that these matrices commute under multiplication. The approach is to impose the condition that A be nonderogatory, which allows us to have a basis for the solution space of B. We analyze a type of matrix called a companion matrix, and we will show that the companion matrix commutes with a wider variety of partial matrix patterns than matrices examined in the past.