Classification of matrices using their numerical range

Abstract

A matrix classification problem is defined where the categories are obtained from predefined eigenvalue structures. A straightforward solution for this problem is to compute the eigenvalues of the observed matrix. Let A be the n x n matrix used in this case. When working with noisy data, from which the entries for A are obtained, and when n is large, we deal with ill-conditioned problems in general, i.e., under small perturbations in the data, eigenvalue computations are largely off or even fail to converge. In this work, we propose to approach the classification problem using the numerical range or field of values of A. The numerical range of A is a convex set in the complex plane that contains information about A. In particular, it can be proved, that the eigenvalues of a matrix lie within its numerical range. Now, suppose that p classes of matrices are defined by p candidate sets of complex numbers, each containing the eigenvalues (or a subset of them) of the representative matrix for each class. Now, the problem is to determine which one of the p candidate sets is included in the numerical range of the matrix A. If the i-th set lies into the numerical range of A, we say that A belongs to the i-th class. Then, the classification problem is molded into a particular set inclusion problem. In this work, we present an algorithm to solve the latter without explicit computation of the numerical range. Numerical results are presented and analyzed.