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Derivatives of orthogonal projectors and pseudoinverses were first considered by Golub and Pereyra [378, 1973]Stewart [731, 1977] gives asymptotic forms and derivatives for orthogonal projectors, pseudoinverses, and least squares solutions. 5. Componentwise perturbation analysis. There are several drawbacks with a normwise perturbation analysis. As already mentioned, it can give huge overestimates when the corresponding problem is badly scaled. Using norms we ignore how the perturbation is distributed among the elements of the matrix and vector.
If only part of the Penrose conditions hold, the corresponding matrix X is called a generalized inverse. Such inverses have been extensively analyzed; see Nashed [596, 1976]. The pseudoinverse can be shown to have the following properties. 12. 8. A, AH. At, and At A all have rank equal to trace (At A). Proof. 23). See also Penrose [655, i955]The pseudoinverse does not share some other properties of the ordinary inverse. 2. 2. 3 and relates to the least squares solution in the case of full column rank.
16) Proof. The result is established in almost the same way as for the corresponding eigenvalue theorem, the Courant-Fischer theorem; see Wilkinson [836, 1965, pp. 99-101]. The minmax characterization of the singular values may be used to establish results on the sensitivity of the singular values of A to perturbations. 7. 18) Proof. See Stewart [729, 1973, pp. 321-322]. 18) is known as the Wielandt-Hoffman theorem for singular values. The theorem shows the important fact that the singular values of a matrix A are well-conditioned with respect to perturbations of A.