Frames of orthogonal projections

grantor: University of TorontoThe primary aim of this thesis is to find and compare appropriate notions of distances on frames which arise from different contexts. A frame [epsilon] is a collection {'E'1, ... , ' Er'} of mutually orthogonal projections in 'M n' whose sum is the identity matrix 'I'. A frame may be identified with the pinching operator A i=1r 'EiAEi' in B ('Mn'), or with the coset of a certain subgroup of 'Un' (i.e., as a point in a generalized flag manifold). Angles, analogous to those between a pair of subspaces (equivalently, projections), are defined between a pair of frames to measure the distance between them: [Theta] is precisely the set of canonical angles between two pinchings (considered as projections in B ('Mn')), and [Phi]0, [Phi] + are derived from the union of the canonical angles between the constituent projections of two frames. Norm inequalities in both directions are found between these sets of angles. By viewing a frame as a coset, and the arguments of the spectrum of a unitary representative for the coset as angles, some additional relations are derived. The question of when frames are antipodal with respect to a certain natural metric is also addressed; results are obtained for dimensions 'n' = 2, 3, 4.Ph.D