Numerical methods for computing angles between linear subspaces
Åke Björck, Gene Howard Golub · Mathematics of Computation · 1973
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 ) {\theta _k}(F,G) and orthogonal sets of principal vectors u k ∈ F {u_k} \in F and v k ∈ G , k = 1 , 2 , ⋯ , q = dim ( G ) ≦ dim ( F ) {v_k} \in G,k = 1,2, \cdots ,q = \dim (G) \leqq \dim (F) . An important application in statistics is computing the canonical correlations σ k = cos θ k {\sigma _k} = \cos {\theta _k} between two sets of variates. A perturbation analysis shows that the condition number for θ k {\theta _k} essentially is max ( κ ( A ) , κ ( B ) ) \max (\kappa (A),\kappa (B)) , where κ