Research Projects

Matrices and operators are ubiquitous throughout science, engineering, and mathematics; they are the transformations that arise whenever one studies a linear system (or approximates a nonlinear system by a linear one). Examples include rotations and reflections (rigid motions of space), spin operators (quantum mechanics and quantum computing), stress tensors (mechanics), regression and curve fitting (statistics and data analysis), derivatives and linear differential operators (dynamical systems), to name just a few.  By studying various properties, relations, and transformations of matrices and operators one may obtain insight into a wide range of phenomena.

One particular class of problems of interest is the study of preservers.  For example, if M_n denotes the space of n x n matrices, one might ask for a complete classification of the isometries preserving a fixed norm.  More generally, given any (possibly multi-valued) function f of a matrix (such as its determinant, rank, eigenvalues, singular values, numerical range, etc) one can ask for a description of the maps T:M_n -> M_n satisfying f(T(A)) = f(A) for all A in M_n; in this case one says that T preserves f.  Usually one imposes some additional structure on T, requiring that it be linear, or simply additive, or multiplicative, and so on.  One might also wish to describe those maps T leaving certain special subsets of matrices invariant (such as projections, unitaries, rank one matrices, etc.).  A broad range of tools and concepts are used in solving such preserver problems; for example, consideration of the dual norm, coupled with convexity arguments, can be handy in classifying isometries, while majorization may appear in problems involving eigenvalues, singular values, and unitarily invariant norms.  Currently, investigation is being conducted on isometries of certain matrix subalgebras, as well as preservers of certain collections of projections.