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.