A geometric approach to subspace updates and orthogonal matrix decompositions under rank-one modifications

For a matrix X ∈ R^n×p, we provide an analytic formula that keeps track of an orthonormal basis for the range of X under rank-one modifications. More precisely, we consider rank-one adaptations X_new = X + ab^T of a given X with known matrix factorization X = UW, where U ∈ R^n×p is column-orthogonal and W ∈ R^p×p is invertible. Arguably, the most important methods that produce such factorizations are the singular value decomposition (SVD), where X = UW = U(ΣV ^T), and the QR-decomposition, where X = UW = QR. We give a geometric description of rank-one adaptations and derive a closed-form expression for the geodesic line that travels from the subspace S = ran(X) to the subspace S_new = ran(X_new) = ran(U_newW_new). This leads to update formulas for orthogonal matrix decompositions, where both U_new and W_new are obtained via elementary rank-one matrix updates in O(np) time for n » p. Moreover, this allows us to determine the subspace distance and the Riemannian midpoint between the subspaces S and S_new without additional computational effort.