Canonical angles and limits of sequences of EP and co-EP matrices

Let A be a square complex matrix. Let P be one of the following properties: (a) A is an EP matrix, (b) the column space of A is complementary to the column space of A Ã , and (c) the orthogonal complement of the column space of A is the column space of A Ã. We study the canonical angles between the column space of A and the column space of A Ã when A satisfies property P. Also, we research the following problem: Let fA m g 1 m¼1 be a sequence of matrices satisfying property P that converges to some matrix A. When does A satisfy property P? The canonical angles (also called principal angles) between two subspaces provide the best available characterization of the relative position of two given subspaces. This concept allows us to characterize or measure, in a natural way, how two subspaces differ, which is the main connection with perturbation theory. In [1–3] we can find how these angles were discovered and rediscovered again several times. Computation of canonical angles between subspaces is important in many applications including statistics [4,5], information retrieval [6], and analysis of algorithms [7]. There are many equivalent definitions of the canonical angles (see [8]). For our purposes, the most convenient is the following: