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 ) {θ _k}(F,G) and orthogonal sets of principal vectors u k ∈ F {u_k} ∈ F and v k ∈ G , k = 1 , 2 , ⋯ , q = dim ⁡ ( G ) ≦ dim ⁡ ( F ) {v_k} ∈ G,k = 1,2, ⋯ ,q = (G) (F) . An important application in statistics is computing the canonical correlations σ k = cos ⁡ θ k {σ _k} = cos {θ _k} between two sets of variates. A perturbation analysis shows that the condition number for θ k {θ _k} essentially is max ( κ ( A ) , κ ( B ) ) max (κ (A),κ (B)) , where κ