New bounds for perturbation of the orthogonal projection
For a matrix \(M\) with Moore-Penrose inverse \(M^\dag\), define the projection matrix \(P_M=MM^\dag\). Consider a matrix \(A\) and its perturbation \(B\) that can be additive (\(B=A+E\)) or multiplicative (\(B=D_1AD_2\)) then this paper gives a bound for \(\|P_A-P_B\|\) for a general unitarily invariant norm. The analysis is based on the singular value decomposition of the matrix \(A\) and is a slight improvement over known estimates. The disadvantage is however that the bounds not only assume some knowledge of the perturbing matrices and the singular values of \(A\) but they also involve the singular vectors.
scientific article; zbMATH DE number 6153976