The Angle Between Complementary Subspaces

Abstract

Usually the discussion stops right there, and extensions to angles between subspaces of higher dimensions are, more or less tacitly, shoved under the rug. Perhaps this is because most instructors feel that such extensions are difficult to understand, or that further effort in this direction is not worthwhile. Indeed, this makes sense for angles between general subspaces because one would have to introduce concepts like gap or distance between subspaces [7, 12], principal (or canonical) angles [1, 2, 15, 12], the CS decomposition [11, 4, 10, 6, 12], and so on. These topics are better off in a more advanced course. However, angles between complementary subspaces are easier to deal with. The purpose of our article is to draw attention to some simple, though not very well known, expressions for the angle between complementary subspaces which are easily derived from the fundamental theorem of linear algebra [14] and elementary facts about matrix norms and projectors. Angles between complementary subspaces are not just academic. They arise, for instance, in the context of controller robustness [9, 16]. Roughly speaking, the spaces associated with the controller and the plant (a system described by a set of differential equations) are complementary subspaces. The robustness of the controller is defined by the smallest perturbation that renders the system unstable, which means that the associated subspaces are no longer complementary. The system remains stable as long as perturbations are smaller than the distance between the complementary subspaces. One measure of distance is the sine of the angle between the spaces.