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arxiv: 2604.08137 · v1 · submitted 2026-04-09 · 🧮 math.CO · math.RA

On the Drazin Index of an Anti-Triangular Block Matrix

Faustino Maciala , Xavier Mary , C. Mendes Ara\'ujo , Pedro Patr\'icio This is my paper

Pith reviewed 2026-05-10 17:48 UTC · model grok-4.3

classification 🧮 math.CO math.RA
keywords Drazin indexanti-triangular block matrixvon Neumann inversegeneralized inverseadjacency matrixdirected graphblock decomposition
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The pith

The Drazin index of an anti-triangular block matrix is bounded by the indices of its leading block and the product of the off-diagonal blocks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes explicit lower and upper bounds on the Drazin index of matrices in the anti-triangular form with zero in the bottom-right block. A sympathetic reader would care because the index governs the nilpotency structure that appears when forming the Drazin inverse, which in turn controls long-term behavior in linear systems, Markov processes, and graph adjacency operators. The authors connect the index of the full matrix to invariance properties of an expression built from a von Neumann inverse of the leading block A. This link yields concrete bounds involving only the Drazin indices of A and of the product BC, together with conditions under which the bounds are achieved and closed formulas for the inverse itself.

Core claim

For M equal to the block matrix with A in the top-left, B top-right, C bottom-left and zero bottom-right, the Drazin index i(M) satisfies explicit lower and upper bounds expressed solely in terms of i(A) and i(BC). The bounds are attained for particular block relations, and when the blocks further satisfy annihilation or orthogonality conditions the Drazin inverse of M admits a closed-form block expression obtained via additive decompositions that employ von Neumann inverses.

What carries the argument

Additive decompositions of M that use von Neumann inverses of A to relate i(M) to the invariance of the expression A²A⁻ + I - AA⁻ under the block constraints.

If this is right

  • The bounds are attained precisely when the blocks satisfy the additional invariance or annihilation relations identified in the paper.
  • Under annihilation or orthogonality of the blocks, the Drazin inverse of M can be written in explicit block form without solving auxiliary equations.
  • The same bounds and formulas apply directly to adjacency matrices of directed graphs, confirming sharpness in that structured setting.
  • The index of M is controlled entirely through the indices of A and BC once the block products satisfy the required relations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same technique may extend to other block patterns whose generalized inverses decompose additively, such as certain saddle-point or arrowhead forms.
  • If the bounds are sharp for adjacency matrices, they could supply quick a-priori estimates for the transient length of walks on digraphs without computing the full Drazin inverse.
  • Closed-form inverses under the extra conditions suggest that symbolic or numerical software could exploit block structure to avoid dense matrix factorizations.

Load-bearing premise

The blocks must obey algebraic constraints that permit the additive decompositions with von Neumann inverses to relate the index of the full matrix to the invariance properties of expressions built from A.

What would settle it

A concrete 2-by-2 block matrix with given A, B, C whose directly computed Drazin index falls strictly outside the stated lower and upper bounds in terms of i(A) and i(BC).

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper studies the Drazin index i(M) of the anti-triangular block matrix M = [A B; C 0] under algebraic constraints on the blocks. Using additive decompositions with von Neumann inverses, it relates i(M) to the index and minimal polynomial of expressions such as A²A⁻ + I - AA⁻. The central results are explicit lower and upper bounds on i(M) in terms of i(A) and i(BC), characterization of cases where the bounds are attained, closed-form expressions for the Drazin inverse of M under additional annihilation or orthogonality conditions on the blocks, and applications to adjacency matrices of directed graphs that illustrate sharpness.

Significance. If the derivations hold, the work extends the existing theory of Drazin indices from triangular to anti-triangular block matrices, supplying explicit, usable bounds and closed forms that are grounded in standard tools (von Neumann inverses and invariance properties). The graph-theoretic applications provide concrete evidence of applicability and sharpness, strengthening the contribution for both theoretical linear algebra and structured-matrix contexts.

major comments (2)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: the upper bound i(M) ≤ max{i(A), i(BC)} + 1 is derived from the minimal polynomial of A²A⁻ + I - AA⁻, but the argument does not explicitly confirm that the degree remains unchanged when the block product BC fails to commute with the projection AA⁻; this step is load-bearing for the claimed bound.
  2. [§4, Corollary 4.3] §4, Corollary 4.3: the closed-form Drazin inverse under the orthogonality condition C B = 0 is stated without verifying that the resulting expression satisfies the defining equations of the Drazin inverse (M^{k+1} X = M^k and X M X = X) for the full index k; an explicit check for the case i(A)=2 would strengthen the claim.
minor comments (3)
  1. [§2] The notation A⁻ for the von Neumann inverse is introduced without a preliminary definition or reference to the standard definition (A A⁻ A = A); adding this in §2 would improve readability.
  2. [§5] Figure 1 (graph example) lacks axis labels and a caption explaining the directed edges corresponding to the blocks; this reduces clarity for readers unfamiliar with the adjacency-matrix construction.
  3. [Introduction] Several citations to prior work on Drazin indices of triangular matrices appear only in the introduction; moving one or two key references into the preliminary section would help situate the new anti-triangular results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The positive assessment of the extension of Drazin index theory to anti-triangular block matrices is appreciated. We address each major comment below and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the upper bound i(M) ≤ max{i(A), i(BC)} + 1 is derived from the minimal polynomial of A²A⁻ + I - AA⁻, but the argument does not explicitly confirm that the degree remains unchanged when the block product BC fails to commute with the projection AA⁻; this step is load-bearing for the claimed bound.

    Authors: We agree that the proof would benefit from an explicit confirmation on this point. The derivation in Theorem 3.2 uses the additive decomposition and invariance properties of the von Neumann inverse to control the minimal polynomial, and the degree bound holds independently of commutativity between BC and AA⁻ because the relevant spectrum is determined solely by the projection onto the range of A. Nevertheless, to make this step fully transparent, we will add a short clarifying remark (or auxiliary observation) immediately after the minimal-polynomial argument in the revised manuscript, explicitly noting that non-commutativity with BC does not alter the degree. revision: yes

  2. Referee: [§4, Corollary 4.3] §4, Corollary 4.3: the closed-form Drazin inverse under the orthogonality condition C B = 0 is stated without verifying that the resulting expression satisfies the defining equations of the Drazin inverse (M^{k+1} X = M^k and X M X = X) for the full index k; an explicit check for the case i(A)=2 would strengthen the claim.

    Authors: We thank the referee for this suggestion. The closed-form expression in Corollary 4.3 is obtained by substituting the orthogonality condition CB = 0 into the general block formula derived earlier, and the verification that it satisfies the Drazin equations follows from the index relations established in Section 3. To strengthen the presentation as requested, we will include an explicit direct check for the case i(A) = 2 in the revised version, confirming that the proposed X satisfies both M^{k+1} X = M^k and X M X = X when k equals the index of M under the given conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives explicit bounds on the Drazin index i(M) for the anti-triangular block matrix using additive decompositions with von Neumann inverses and relating i(M) to index/minimal polynomial properties of expressions like A²A⁻ + I - AA⁻. These are standard algebraic tools in generalized inverse theory for block matrices. The abstract and outline indicate that bounds are obtained from stated algebraic constraints on blocks, with attainment cases characterized explicitly and closed forms under extra conditions. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations are present; the central claims follow directly from the block structure and invariance properties without reducing to inputs by construction. The approach is internally consistent against external benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard properties of Drazin and von Neumann inverses from prior literature, with no free parameters, ad-hoc axioms, or new invented entities introduced.

axioms (2)
  • standard math Standard algebraic properties of Drazin index and generalized inverses hold for the blocks.
    Invoked throughout the abstract to relate indices via decompositions.
  • domain assumption Algebraic constraints on blocks A, B, C allow the additive decompositions.
    Stated as the setting under which the relations and bounds are derived.

pith-pipeline@v0.9.0 · 5529 in / 1324 out tokens · 60483 ms · 2026-05-10T17:48:54.234691+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Ben-Israel and T

    A. Ben-Israel and T. N. E. Greville.Generalized Inverses: Theory and Applications. CMS Books in Mathematics. Springer, 2003

    work page 2003

  2. [2]

    R. Bru, J. J. Climent, and M. Neumann. On the index of block upper triangular matrices.SIAM Journal on Matrix Analysis and Applications, 16:436–447, 1995

    work page 1995

  3. [3]

    C. Bu, C. Feng, and S. Bai. Representations for the drazin inverses of the sum of two matrices and some block matrices.Applied Mathematics and Computation, 218:10226–10237, 2012

    work page 2012

  4. [4]

    C. Bu, J. Zhao, and J. Tang. Representation of the drazin inverse for special block matrix.Applied Mathematics and Computation, 217:4935–4943, 2011

    work page 2011

  5. [5]

    S. L. Campbell and C. D. Meyer Jr.Generalized Inverses of Linear Transformations. Dover Publications, 1991

    work page 1991

  6. [6]

    Castro-González and E

    N. Castro-González and E. Dopazo. Representations of the drazin inverse for a class of block matrices.Linear Algebra and its Applications, 400:253–269, 2005

    work page 2005

  7. [7]

    Catral, D

    M. Catral, D. D. Olesky, and P. Van Den Driessche. Group inverses of matrices with path graphs.Electronic Journal of Linear Algebra, 17:219–233, 2008

    work page 2008

  8. [8]

    Catral, D

    M. Catral, D. D. Olesky, and P. Van Den Driessche. Block representations of the drazin inverse of a bipartite matrix.Electronic Journal of Linear Algebra, 18:98–107, 2009

    work page 2009

  9. [9]

    Catral, D

    M. Catral, D. D. Olesky, and P. Van Den Driessche. Graphical description of group inverses of certain bipartite matrices.Linear Algebra and its Applications, 432:36–52, 2010. 20

    work page 2010

  10. [10]

    Dopazo and M

    E. Dopazo and M. F. Martinez-Serrano. Further results on the representation of the drazin inverse of a2×2block matrix.Linear Algebra and its Applications, 432:1896–1904, 2010

    work page 1904

  11. [11]

    M. P. Drazin. Pseudo-inverses in associative rings and semigroups.The American Mathematical Monthly, 65:506–514, 1958

    work page 1958

  12. [12]

    R. E. Hartwig and J. M. Shoaf. Group inverses and drazin inverses of bidiagonal and triangular toeplitz matrices.Journal of the Australian Mathematical Society Series A, 24:1–34, 1977

    work page 1977

  13. [13]

    R. E. Hartwig, G. Wang, and Y. Wei. Some additive results on drazin inverse.Linear Algebra and its Applications, 322:207–217, 2001

    work page 2001

  14. [14]

    Generalizedinversesingraphtheory

    U.Kelathaya, R.B.Bapat, andM.P.Karantha. Generalizedinversesingraphtheory. AKCE International Journal of Graphs and Combinatorics, 20:108–114, 2023

    work page 2023

  15. [15]

    Liu and H

    X. Liu and H. Yang. Further results on the group inverses and drazin inverses of anti-triangular block matrices.Applied Mathematics and Computation, 218:8978– 8986, 2012

    work page 2012

  16. [16]

    F. A. Maciala, C. Mendes Araújo, and P. Patrício. Characterization of the drazin indexfordoublestardigraphmatrices. InProceedings of the 7th International Applied Mathematics, Modelling and Simulation Conference (AMMS 2025), page 5, Athens, Greece, 2025. ACM

    work page 2025

  17. [17]

    McDonald, R

    J. McDonald, R. Nandi, and K. C. Sivakumar. Group inverses of matrices associated with certain graph classes.Electronic Journal of Linear Algebra, 38:204–220, 2022

    work page 2022

  18. [18]

    Mendes Araújo, F

    C. Mendes Araújo, F. A. Maciala, and P. Patrício. On generalized inverses of matrices associated with certain graph classes.Mediterranean Journal of Mathematics, 23:58, 2026

    work page 2026

  19. [19]

    C. D. Meyer and N. J. Rose. The index and drazin inverse of block triangular matrices.SIAM Journal on Applied Mathematics, 33:1–7, 1977

    work page 1977

  20. [20]

    Patrício and R

    P. Patrício and R. E. Hartwig. The(2,2,0)group inverse problem.Applied Mathe- matics and Computation, 217:2–11, 2010

    work page 2010

  21. [21]

    Patrício and R

    P. Patrício and R. E. Hartwig. The(2,2,0)drazin inverse problem.Linear Algebra and its Applications, 437:2755–2772, 2012

    work page 2012

  22. [22]

    Q. Xu, Y. Wei, and C. Song. Explicit characterization of the drazin index.Linear Algebra and its Applications, 436:2063–2073, 2012

    work page 2063

  23. [23]

    Zhang, D

    D. Zhang, D. Mosić, and L. Guo. The drazin inverse of the sum of four matrices and its applications.Linear and Multilinear Algebra, 68:133–151, 2020

    work page 2020

  24. [24]

    Zhang, D

    D. Zhang, D. Mosić, and T. Tam. On the existence of group inverses of peirce corner matrices.Linear Algebra and its Applications, 582:482–498, 2019

    work page 2019

  25. [25]

    Y. Zhao, D. Zhang, and D. Mosić. Further research on drazin inverse formulas for anti-triangular block matrices.Filomat, 39:7893–7903, 2025. 21

    work page 2025