PRINCIPAL ANGLES AND APPROXIMATION 177

This paper is dedicated to Professor Tsuyoshi Ando, in celebration of his expertise in matrix and operator theory Communicated by G. Androulakis Abstract. We extend Jordan’s notion of principal angles to work for two sub-spaces of quaternionic space, and so have a method to analyze two orthogonal projections in the matrices over the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real C∗-algebra generated by two projections. 1. Two projections, the three-fold way The general form of two projections on complex Hilbert space is well-known, going back to at least Dixmier [6]. The real case is older, being implicit in the work of Jordan [13, §IV], where principal vectors and principal angles were introduced. From principal vectors one can derive the structure theorem for matrix projections, as is explained in the real case in [8]. Restricted to the finite-dimensional case, we can think of these as theorems about two projections in certain finite-dimensional real C∗-algebras. One would therefore expect the same result to hold in all finite-dimensional real C∗-algebras, and so in Mn(H) where H is the skew field of quaternions