PRODUCT COSINES OF ANGLES BETWEEN SUBSPACES

Let cos{L, M} = Pi(i=1) cos theta(i) denote the product of the cosines of the principal angles (theta(i)) between the subspaces L and M. The direction cosines of an r-dimensional subspace L are the [GRAPHICS] n umbers {cos{L,R(j)(n)} : J is an element of Q(r,n)}, where Q(r,n) = th e set of increasing sequences of r elements from {1,..., n}, and R(j)( n) = {x = (x(k)) is an element of R(n) : x(k) = 0 for k is not an elem ent of J}. The basic decomposition of a linear operator A : R(n) --> R (m), with rank A = r > 0, is [GRAPHICS] a convex combination of nonsin gular linear operators B-IJ : R(J)(n) --> R(I)(n). Here J(A) = {I is a n element of Q(r,m) : rank A(l) = r} and J(A) = {J is an element of Q (r,n) : rank A(J) = r}. The product cosines are related to the matrix volume, defined as the product of its nonzero singular values. The Mo ore-Penrose inverse A(dagger) is characterized as having the minimal v olume among all {1, 2}-inverses of A. Indeed, if G is a {1, 2}-inverse of A, with range R(G) = T and null space N(G) = S, then vol G = vol A (dagger)/cos{T,R(A(T))}cos{S,N(A(T))}.