On the Drazin Index of an Anti-Triangular Block Matrix

TL;DR

This paper investigates the Drazin index of anti-triangular block matrices, establishing bounds, invariance properties, and explicit formulas under certain conditions, with applications to graph adjacency matrices.

Contribution

It extends the theory of Drazin indices to anti-triangular matrices, providing bounds, characterizations, and inverse formulas under algebraic constraints.

Findings

Derived explicit bounds for the Drazin index in terms of block indices.

Characterized conditions for when bounds are attained.

Provided closed-form Drazin inverse formulas under additional conditions.

Abstract

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 \[ M=\begin{bmatrix} A & B \\ C & 0 \end{bmatrix}, \] 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 $A^{2}A^{-}+I-AA^{-}$. 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.