Acute perturbation of Drazin inverse and oblique projectors (Q1731906)

For an $n\times n$ complex matrix $A$ with $\text{ind}(A) = r$, let $A^D$ and $A^\pi = I- AA^D$ be the Drazin inverse and the eigenprojection corresponding to the eigenvalue $0$ of $A$, respectively. An $n\times n$ matrix $B$ is said to be an acute perturbation of $A$ with respect to the Drazin inverse if $\rho(B^\pi -A^\pi) < 1$, where $\rho$ denotes the spectral radius. The authors give necessary and sufficient conditions for a perturbation being acute with respect to the Drazin inverse and thus generalize the corresponding results for the group inverse; see [\textit{Y. Wei}, Linear Algebra Appl. 534, 135--157 (2017; Zbl 1371.15005)]. In addition, they follow [\textit{N. Castro-González} et al., SIAM J. Matrix Anal. Appl. 30, No. 2, 882--897 (2008; Zbl 1165.15004)] to derive new formulae for the spectral radii $\rho((I-B^\pi)A^\pi)$ and $\rho((I-A^\pi)B^\pi)$ under certain conditions. They also generalize the perturbation analysis to oblique projectors and present some applications.