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 null spaces) of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles θ k (F, G) ₖ (F, G) and orthogonal sets of principal vectors u k ∈ F uₖ F and v k ∈ G, k = 1, 2, ⋯, q = dim ⁡ (G) ≦ dim ⁡ (F) vₖ G, k = 1, 2, , q = (G) (F). An important application in statistics is computing the canonical correlations σ k = cos ⁡ θ k ₖ = ₖ between two sets of variates. A perturbation analysis shows that the condition number for θ k ₖ essentially is max (κ (A), κ (B) ) ( (A), (B) ), where κ