A closed-form update for orthogonal matrix decompositions under arbitrary rank-one modifications

We consider rank-one adaptations $X_{new} = X+ab^T$ of a given matrix $X\in \mathbb{R}^{n\times p}$ with known matrix factorization $X = UW$, where $U\in\mathbb{R}^{n\times p}$ is column-orthogonal, i.e. $U^TU=I$. Arguably the most important methods that produce such factorizations are the singular value decomposition (SVD), where $X=UW=U\Sigma V^T$, and the QR-decomposition, where $X = UW = QR$. By using a geometric approach, we derive a closed-form expression for a column-orthogonal matrix $U_{new}$ whose columns span the same subspace as the columns of the rank-one modified $X_{new} = X +ab^T$. This may be interpreted as a rank-one adaptation of the $U$-factor in the SVD or a rank-one adaptation of the $Q$-factor in the QR-decomposition, respectively. As a consequence, we obtain a decomposition for the adapted matrix $X_{new} = U_{new}W_{new}$. Moreover, the formula for $U_{new}$ allows us to determine the subspace distance between the subspaces colspan$(X) =\mathcal{S}$ and colspan$(X_{new}) =\mathcal{S}_{new}$ without additional computational effort. In contrast to the existing approaches, the method does not require a numerical recomputation of the SVD or the QR-decomposition of an auxiliary matrix as an intermediate step.