Perturbation bounds for the generalized inverses \(A_{T,S}^{(2)}\) with application to constrained linear system.
Under suitable conditions, perturbation bounds for the \(\{2\}\)-inverse matrix \(A_{T,S}^{(2)}\) with prescribed range \(T\) and null space \(S\) are derived when the matrix \(A\) and the subspaces \(T\) and \(S\) are perturbed. In particular a result of \textit{Y. Wei} and \textit{H. Wu} [J. Comput. Appl. Math. 137, No. 2, 317--329 (2001; Zbl 0993.15003)] is generalized, and the continuity result of \(A_{T,S}^{(2)}\) is easily derived. Then condition numbers of the generalized inverse \(A_{T,S}^{(2)}\) are defined generalizing the definition of the condition number of a nonsingular matrix and methods to compute them are obtained. For example it is proved that \[ \text{Cond}_{\text{F}}^{TS} (A) = \frac{\| A_{T,S}^{(2)}\| _2^2 \| A\| _{\text{F}}}{\| A_{T,S}^{(2)}\| _{\text{F}}},\qquad \text{Cond}_{2}^{TS} (A) = \| A\| _2 \| A_{T,S}^{(2)}\| _2 . \] Under natural assumptions expressions for the condition numbers of the Moore-Penrose and Drazin inverses are obtained substituting in the former equalities the generalized inverse \(A_{T,S}^{(2)}\) by the Moore-Penrose and Drazin inverses respectively. Finally values for the condition numbers of constrained linear systems are obtained, in the general case, and when the Moore-Penrose and Drazin inverses are used.
scientific article; zbMATH DE number 1957138
scientific article