The product field of values

For two n-by-n matrices, A, B, the product field of values is the set P(A, B) = {\u3c AX, X \u3e \u3c Bx, X \u3e : X is an element of C-n, parallel to X parallel to = 1}. In this paper, we establish basic properties of the product field of values. The main results are a proof that the product field is a simply connected subset of the complex plane and a characterization of matrix pairs for which the product field has nonempty interior. (C) 2012 Elsevier Inc. All rights reserved

The ratio field of values

The ratio field of values, a generalization of the classical field of values to a pair of n-by-n matrices, is defined and studied, primarily from a geometric point of view. Basic functional properties of the ratio field are developed and used. A decomposition of the ratio field into line segments and ellipses along a master curve is given and this allows computation. Primary theoretical results include the following. It is shown (1) for which denominator matrices the ratio field is always convex, (2) certain other cases of convex pairs are given, and (3) that, at least for n = 2, the ratio field obeys a near convexity property that the intersection with any line segment has at most n components. Generalizations of the ratio field of values involving more than one matrix in both the numerator and denominator are also investigated. It is shown that generally such extensions need not be convex or even simply connected. (c) 2010 Elsevier Inc. All rights reserved