Numerical methods for computing angles between linear subspaces

摘要: Assume that two subspaces F and G of unitary space are defined as the ranges (or nullspaces) given rectangular matrices A B. Accurate numerical methods developed for computing principal angles $\theta_k (F,G)$ orthogonal sets vectors $u_k\ \epsilon\ F$ $v_k\ G$, k = 1,2,..., q dim(G) $\leq$ dim(F). An important application in statistics is canonical correlations $\sigma_k\ cos \theta_k$ between variates. perturbation analysis shows condition number $\theta_k$ essentially max($\kappa (A),\kappa (B)$), where $\kappa$ denotes a matrix. The algorithms based on preliminary QR-factorization B $A^H$ $B^H$), which either method Householder transformations (HT) or modified Gram-Schmidt (MGS) used. Then sin computed singular values certain related matrices. Experimental results given, indicates MGS gives with equal precision fewer arithmetic operations than HT. However, HT vectors, to working accuracy, not general true MGS. Finally case when and/or rank deficient discussed.