The calculation of the dominant eigenvalues of a symmetric matrix A together with its eigenvectors, followed by the calculation of the deflation of A1=A−ρUkUkT corresponds to one step of the Wielandt deflation technique, where ρ is a shift and Uk are eigenvectors of A. In this paper, we investigate how the eigenspace of A1 changes when A1 is perturbed to A˜1=A−ρU˜kU˜kT, where U˜k are approximate eigenvectors of A. We establish the bounds for the angle of eigenspaces of A1 and 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 B1=B−γWkVkH with γ being a shift, Wk and Vk are singular vectors of B, so that they will not be re-computed. We investigate how the singular subspaces of B1=B−γWkVkH change when B1 is perturbed to B˜1=B−γW˜kV˜kH, W˜k and V˜k are approximate singular vectors of B. We also establish the bounds for the angle of singular subspaces of B1 and B˜1 based on the Wedin theorem.
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