On the index of an anti-triangular block matrix
Abstract
The index of a matrix 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 = [ 𝐴 𝐵 𝐶 0 ] , under algebraic constraints on the blocks. Building on additive decompositions involving von Neumann inverses, we relate the index of M to invariance properties of the index and minimal polynomial of expressions of the form 𝐴 2 𝐴 − + 𝐼 − 𝐴 𝐴 − , where 𝐴 − is any von Neumann inverse of 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 the index of M in terms of the index of A and of 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.
Related
- A Note on the Drazin Inverse of 2 × 2 Block Triangular Matrices
- Randomized inexact block triangular preconditioners for double saddle-point systems in PDE-constrained optimization
- One-Step Generative Surrogate Models via Block-Triangular Joint Drifting
- Limits of Stochastic Semigroups and Block-Triangular Majorisation
- Arithmetic triangular structures in the transfer-matrix of finite Kronig-Penney models
BibTeX
@article{maciala2026index,
title = {{On the index of an anti-triangular block matrix}},
author = {Faustino Maciala and Xavier Mary and C. Mendes Araújo and Pedro Patrício},
journal = {Applied Mathematics and Computation},
year = {2026},
month = oct,
doi = {10.1016/j.amc.2026.130346},
eprint = {2604.08137},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05828531v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - On the index of an anti-triangular block matrix
AU - Faustino Maciala
AU - Xavier Mary
AU - C. Mendes Araújo
AU - Pedro Patrício
PY - 2026
DA - 2026/10/21
JO - Applied Mathematics and Computation
DO - 10.1016/j.amc.2026.130346
UR - https://hal.science/hal-05828531v1
AB - The index of a matrix 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 = [ 𝐴 𝐵 𝐶 0 ] , under algebraic constraints on the blocks. Building on additive decompositions involving von Neumann inverses, we relate the index of M to invariance properties of the index and minimal polynomial of expressions of the form 𝐴 2 𝐴 − + 𝐼 − 𝐴 𝐴 − , where 𝐴 − is any von Neumann inverse of 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 the index of M in terms of the index of A and of 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.
SN - 0096-3003
ER -