\input zb-basic
\input zb-ioport
\iteman{io-port 05811005}
\itemau{Wei, Musheng; Ling, Sitao}
\itemti{On the perturbation bounds of g-inverses and oblique projections.}
\itemso{Linear Algebra Appl. 433, No. 11-12, 1778-1792 (2010).}
\itemab
Given a matrix $A$, a g-inverse (sometimes also called $(1)$-inverse) of $A$ is any matrix $X$ satisfying the relation $AXA = A$. Given a g-inverse $A^-$ of A and a perturbed matrix $\hat A$, the authors investigate the problem of finding a g-inverse of $\hat A$ that is closest to $A^-$ in a certain norm. In the matrix $2$-norm, it turns out that this problem has a unique solution, which can be expressed in terms of the Moore-Penrose pseudoinverse of $\hat A$. In the Frobenius norm, a set of parametrized solutions is given. A variation of this problem, where the aim is to minimize the difference between the corresponding oblique projections onto the range of $A$ and $\hat A$, is also addressed. Extending these ideas, the authors provide perturbation results for solutions of linear least-squares problems with multiple right-hand sides.
\itemrv{Daniel Kressner (Berlin)}
\itemcc{}
\itemut{g-inverse; oblique projection; linear system; perturbation bound; rank preserving; $(1)$-inverse; Moore-Penrose pseudoinverse; linear least-squares problems}
\itemli{doi:10.1016/j.laa.2010.06.036}
\end