On the Drazin Index of an Anti-Triangular Block Matrix

The Drazin index is a fundamental invariant in the analysis of singular matrices and their generalized inverses. While sharp results are available for block triangular matrices, the corresponding theory for anti-triangular block matrices is less developed. In this paper, we study matrices of the form $$M=\left[\begin{array}{cc}A& B\\ C& 0\end{array}\right],$$ under algebraic constraints on the blocks. Building on additive decompositions involving von Neumann inverses, we relate the Drazin index of $M$ to invariance properties of the index and minimal polynomial of expressions of the form $A^2{A}^{-}+I-A{A}^{-}$. This connection provides an effective mechanism to control the index of $M$ through suitable factorizations and associated block products. As a consequence, we derive explicit lower and upper bounds for $i(M)$ in terms of $i(A)$ and $i(BC)$, and characterize situations in which these bounds are attained. Under additional annihilation or orthogonality conditions on the blocks, we obtain closed-form representations for the Drazin inverse of $M$. Applications to adjacency matrices of directed graphs illustrate the sharpness of the bounds and the applicability of the results to structured matrices arising in graph-theoretic settings.