On perturbations for spectrum and singular value decompositions followed by deflation techniques

The calculation of the dominant eigenvalues of a symmetric matrix $A$ together with its eigenvectors, followed by the calculation of the deflation of $A_1=A-\rho U_k{U}_k^T$ corresponds to one step of the Wielandt deflation technique, where $\rho$ is a shift and $U_k$ are eigenvectors of $A$. In this paper, we investigate how the eigenspace of $A_1$ changes when $A_1$ is perturbed to ${\tilde{A}}_1=A-\rho {\tilde{U}}_k{\tilde{U}}_k^T$, where ${\tilde{U}}_k$ are approximate eigenvectors of $A$. We establish the bounds for the angle of eigenspaces of $A_1$ and ${\tilde{A}}_1$ based on the Davis-Kahan theorem. Moreover, in the practical implementation for singular value decomposition, once one or several singular triplets converge to a preset accuracy, they should be deflated by $B_1=B-\gamma W_k{V}_k^H$ with $\gamma$ being a shift, $W_k$ and $V_k$ are singular vectors of $B$, so that they will not be re-computed. We investigate how the singular subspaces of $B_1=B-\gamma W_k{V}_k^H$ change when $B_1$ is perturbed to ${\tilde{B}}_1=B-\gamma {\tilde{W}}_k{\tilde{V}}_k^H$, ${\tilde{W}}_k$ and ${\tilde{V}}_k$ are approximate singular vectors of $B$. We also establish the bounds for the angle of singular subspaces of $B_1$ and ${\tilde{B}}_1$ based on the Wedin theorem.