NC ball maps and changes of variables

In this dissertation, we analyze problems set in a dimension-free, noncommutative setting. To be more specific, we use a class of functions defined by power series in noncommutative variables and evaluate these functions on sets of matrices of all sizes - hence the dimension-free term. These types of functions have recently been used in the study of dimension-free linear system engineering problems [HP07], [CHSY03]. Here we analyze a class of functions called NC analytic with the intention of understanding changes of variables in dimension-free classes of problems in matrix variables. To this end, we force geometric constraints on our analytic functions and ask how this affects the algebraic structure of the series defining them. In particular, we present a characterization of maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary. We study this problem in various cases restricting the variables (and matrices) to have additional symmetric structure and in cases where the variables (and matrices) have no such restrictions. These characterizations are then used to study the more general question of understanding when a dimension-free set is bianalytic to a dimension-free ball. In addition to our study of NC analytic functions, we present a result on a representation of noncommutative rational expressions. Recently there have been studies linking convexity of noncommutative rational functions to linear matrix inequalities, LMIs [HMV06]. The algorithm presented in the final chapter of the thesis presents a necessary step to automatically converting inequalities involving convex rational expressions into LMI