论文信息 - Perturbation theory for pseudo-inverses

Perturbation theory for pseudo-inverses

A perturbation theory for pseudo-inverses is developed. The theory is based on a useful decomposition (theorem 2.1) ofB+ -A+ whereB andA arem ×n matrices. Sharp estimates of ∥B+ -A+∥ are derived for unitary invariant norms whenA andB are of the same rank and ∥B -A∥ is small. Under similar conditions the perturbation of a linear systemAx=b is studied. Realistic bounds on the perturbation ofx=A+b andr=b=Ax are given. Finally it is seen thatA+ andB+ can be compared if and only ifR(A) andR(B) as well asR(AH) andR(BH) are in the acute case. Some theorems valid only in the acute case are also proved.

P. Wedin

[1] S. Afriat. Orthogonal and oblique projectors and the characteristics of pairs of vector spaces , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[2] L. Mirsky. SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS , 1960 .

[3] C. Desoer,et al. A Note on Pseudoinverses , 1963 .

[4] Alston S. Householder,et al. The Theory of Matrices in Numerical Analysis , 1964 .

[5] J. H. Wilkinson,et al. Note on the iterative refinement of least squares solution , 1966 .

[6] Adi Ben-Israel,et al. On Error Bounds for Generalized Inverses , 1966 .

[7] Tosio Kato. Perturbation theory for linear operators , 1966 .

[8] Åke Björck. Iterative refinement of linear least squares solutions I , 1967 .

[9] Å. Björck. Solving linear least squares problems by Gram-Schmidt orthogonalization , 1967 .

[10] G. Stewart. On the Continuity of the Generalized Inverse , 1969 .

[11] C. Lawson,et al. Extensions and applications of the Householder algorithm for solving linear least squares problems , 1969 .

[12] V. Pereyra. Stability of general systems of linear equations , 1969 .

[13] W. Kahan,et al. The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[14] A. Sluis. Stability of solutions of linear algebraic systems , 1970 .

[15] James Hardy Wilkinson,et al. The Least Squares Problem and Pseudo-Inverses , 1970, Comput. J..