Numerical methods for computing angles between linear subspaces

Assume that two subspaces F and G of a unitary space are defined.. as the ranges(or nullspacd of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles ek(F,G) and orthogonal sets of principal vectors u k 6 F and vk c G, k = 1,2,..., q = dim(G) 2 dim(F). An important application in statistics is computing the canonical correlations uk = cos 8 k between two sets of variates. A perturbation analysis shows that the condition number for ek essentially is max(K(A),K(B)), where K denotes the condition number of a matrix. The algorithms are based on a preliminary &R-factorization of A and B (or AH and BH), for which either the method of Householder transformations (HT) or the modified Gram-Schmidt method (MGS) is used. Then cos Ok and sin 0 k are computed as the singular values of certain related matrices. Experimental results are given, which indicates that MGS gives Bk with equal precision and fewer arithmetic operations than HT. However, HT gives principal vectors, which are orthogonal to working accuracy, which is not in general true for MGS. Finally the case when A and/or B are rank deficient is discussed