Minimum Property of Condition Numbers for the Drazin Inverse and Singular Linear Equations

Abstract

For a singular linear equation $Ax=b, {x}\in \mathcal{R}(A^{D})$, a small perturbation matrix $E$ and a vector $\delta b$ are given to $A$ and $b$, respectively.

We then have the perturbed singular linear equation

$(A+E)\widetilde{x}= b+\delta b,\, \widetilde{x}\in \mathcal{R}[(A+E)^{D}]$.

This note is devoted to show the minimum property of the condition numbers on the Drazin inverse $A^{D}$ and the Drazin-inverse solution $A^{D}b$.