Product cosines of angles between subspaces Original Research Article

Let cos{L, M} colon, equals Πi = 1cosθi denote the product of the cosines of the principal angles {θi} between the subspaces L and M. The direction cosines of an r-dimensional subspace L are the image numbers image, where Qr, n colon, equals the set of increasing sequences of r elements from {1, …, n}, and image. The basic decomposition of a linear operator image, with rank A = r> 0, is imageimage(A) imageimage(A) imagecos2{R(A),imageml} imagecos2{R(AT),imagenj}imageBij, a convex combination of nonsingular linear operators image. Here image(A) colon, equals {I set membership, variant Qr, m : rank AI* = r} and image(A) colon, equals {J set membership, variant Qr, n : rank A*J = r}. The product cosines are related to the matrix volume, defined as the product of its nonzero singular values. The Moore-Penrose inverse A† is characterized as having the minimal volume 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 image