Perturbation bounds and properties for the inverse along a matrix

Abstract

Let $A\in\mathbb{C}_{m,n}$ and $D\in\mathbb{C}_{n,m}$. In this paper, we present two different expressions of $A^{\|D}$ by using the singular value decomposition and the Schur decomposition and establish several criteria for the existence of $A^{\|D}$. If $E\in\mathbb{C}_{m,n}$ is a perturbation matrix, we present the bound of $\|(A+E)^{\|D}-A^{\|D}\|_2$. In addition, if $T\in\mathbb{C}_{n,m}$, the perturbation bounds of $\|A^{\|D+T}-A^{\|D}\|_2$ and $\|(A+E)^{\|D+T}-A^{\|D}\|_2$ are given. As a consequence, we give sufficient conditions for the continuity of the inverse along a matrix. Finally, we estimate the upper bounds of $\|B^{\|C}-A^{\|D}\|_F$ provided that $B^{\|C}$ and $A^{\|D}$ both exist.