Decompositions of Periodic Matrices into a Sum of Special Matrices

Esther Garc\'ia; Miguel G\'omez Lozano; Peter Danchev

arxiv: 2404.02777 · v1 · submitted 2024-04-03 · 🧮 math.RA

Decompositions of Periodic Matrices into a Sum of Special Matrices

Peter Danchev , Esther Garc\'ia , Miguel G\'omez Lozano This is my paper

Pith reviewed 2026-05-24 02:30 UTC · model grok-4.3

classification 🧮 math.RA

keywords periodic matricesidempotent matricestorsion matricessquare-zero matricesmatrix decompositionsfield characteristics

0 comments

The pith

Every periodic square matrix over any field decomposes as the sum of an idempotent matrix and a torsion matrix.

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

The authors investigate decompositions of periodic matrices, which satisfy a power equation A^k = A^l for k>l. They prove that over fields of prime characteristic, the rationals, or algebraically closed fields of characteristic zero, any such matrix with rank at least half the dimension can be written as the sum of a square-zero matrix and a torsion matrix. A counterexample shows this fails over the real numbers. Crucially, they establish that over every field, every periodic matrix is the sum of an idempotent matrix and a torsion matrix. This matters because it reveals universal structural properties of periodic matrices in linear algebra.

Core claim

We study the problem of when a periodic square matrix of order n over an arbitrary field F is decomposable into the sum of a square-zero matrix and a torsion matrix, and show that this decomposition can always be obtained for matrices of rank at least n/2 when F is either a field of prime characteristic, or the field of rational numbers, or an algebraically closed field of zero characteristic. We also provide a counterexample to such a decomposition when F equals the field of the real numbers. Moreover, we prove that each periodic square matrix over any field is a sum of an idempotent matrix and a torsion matrix.

What carries the argument

The idempotent-plus-torsion decomposition for periodic matrices over arbitrary fields.

If this is right

Periodic matrices of rank at least n/2 over prime characteristic fields, rationals, or algebraically closed zero-char fields decompose into square-zero plus torsion.
The square-zero plus torsion decomposition fails for some matrices over the real numbers.
The idempotent plus torsion decomposition holds for all periodic matrices over any field without rank restrictions.

Where Pith is reading between the lines

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

Similar decompositions might be investigated for matrices over arbitrary rings rather than fields.
The results could be extended to periodic operators on Banach spaces or other infinite-dimensional settings.
Explicit constructions for small matrices could provide concrete examples of these decompositions.

Load-bearing premise

The notions of periodic, square-zero, torsion, and idempotent matrices are well-defined and the standard linear-algebra operations over the listed fields suffice to guarantee the stated existence of the decompositions.

What would settle it

A periodic square matrix over some field that cannot be expressed as the sum of an idempotent matrix and a torsion matrix would falsify the universal claim.

read the original abstract

We study the problem of when a periodic square matrix of order $n$ over an arbitrary field $\mathbb{F}$ is decomposable into the sum of a square-zero matrix and a torsion matrix, and show that this decomposition can always be obtained for matrices of rank at least $\frac n2$ when $\mathbb{F}$ is either a field of prime characteristic, or the field of rational numbers, or an algebraically closed field of zero characteristic. We also provide a counterexample to such a decomposition when $\mathbb{F}$ equals the field of the real numbers. Moreover, we prove that each periodic square matrix over any field is a sum of an idempotent matrix and a torsion matrix.

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.

Desk Editor's Note private letter to a colleague

The idempotent-plus-torsion decomposition holds for every periodic matrix over any field; the square-zero version is restricted by rank and field with an explicit real counterexample.

read the letter

The main result is that every periodic square matrix over an arbitrary field decomposes as the sum of an idempotent matrix and a torsion matrix. A second result gives conditions for a square-zero plus torsion decomposition when the matrix has rank at least n/2, but only over fields of prime characteristic, the rationals, or algebraically closed fields of characteristic zero; they supply a counterexample showing failure over the reals. Both claims are stated cleanly and the counterexample shows the authors checked where the second claim breaks. The idempotent result stands out for its lack of extra restrictions on the field or the period. The work is narrow but the statements are direct and the field dependence is handled explicitly. The abstract supplies no proof sketches or construction details, so it is impossible to judge whether the arguments are elementary or rely on heavy machinery; that is the main limitation visible from the given text. The paper targets people working on additive decompositions inside matrix rings over fields. A reader who cares about existence results for periodic elements will get something concrete from it. The claims are specific enough and the counterexample is useful enough that the paper should go to a serious referee rather than a desk reject.

Referee Report

0 major / 1 minor

Summary. The paper studies decompositions of periodic square matrices of order n over an arbitrary field F. It shows that matrices of rank at least n/2 admit a decomposition into a square-zero matrix plus a torsion matrix when F has prime characteristic, when F = Q, or when F is algebraically closed of characteristic zero; a counterexample is supplied over the reals. It further proves that every periodic square matrix over any field decomposes as the sum of an idempotent matrix and a torsion matrix.

Significance. If the stated existence results hold, the work supplies field-dependent and uniform decompositions for periodic matrices, with an explicit counterexample demonstrating that the square-zero + torsion decomposition is not universal. The uniform idempotent + torsion result over arbitrary fields would be a clean structural statement in linear algebra over rings and fields.

minor comments (1)

The abstract states the main theorems but the provided text contains no proofs, explicit constructions, or error analysis; the full manuscript must be checked for the details of the existence arguments and the counterexample.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. No specific major comments appear in the report, so there are no individual points requiring a response or revision.

Circularity Check

0 steps flagged

No circularity: direct existence theorems in linear algebra

full rationale

The paper states existence results for matrix decompositions (periodic matrix = idempotent + torsion over any field; conditional results for square-zero + torsion). These are proved via standard linear-algebra constructions over fields, with an explicit counter-example supplied for the real numbers. No equations reduce a claimed prediction to a fitted input, no self-definitional loops appear, and no load-bearing step relies on self-citation chains or imported uniqueness theorems. The derivation chain is self-contained against external benchmarks (field arithmetic and matrix algebra), yielding score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on standard definitions and field axioms of linear algebra; no free parameters, new entities, or ad-hoc assumptions are introduced in the abstract.

axioms (1)

standard math Standard definitions and algebraic properties of matrices over arbitrary fields
The statements rely on the usual notions of matrix addition, multiplication, rank, and the listed field classes.

pith-pipeline@v0.9.0 · 5644 in / 1168 out tokens · 26778 ms · 2026-05-24T02:30:42.704349+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We also prove that each periodic square matrix over any field is a sum of an idempotent matrix and a torsion matrix.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A square matrix A is called periodic if Am = An for certain m > n ≥ 1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

A. N. Abyzov and D. T. Tapkin. Rings over which every matri ces are sums of idempotent and q-potent matrices. Siberian Math. J. , 62(1) (2021) 1–13

work page 2021
[2]

A. N. Abyzov, D. T. Tapkin. When is every matrix over a ring the sum of two tripotents? Linear Algebra Appl. , 630 (2021) 316–325

work page 2021
[3]

S. Breaz. Matrices over ﬁnite ﬁelds as sums of periodic an d nilpotent elements. Linear Algebra Appl., 555 (2018) 92–97

work page 2018
[4]

Breaz, G

S. Breaz, G. Cˇ alugˇ areanu, P. Danchev and T. Micu. Nil-c lean matrix rings. Linear Algebra Appl., 439 (2013) 3115–3119

work page 2013
[5]

Breaz and S

S. Breaz and S. Megiesan. Nonderogatory matrices as sums of idempotent and nilpotent matrices. Linear Algebra Appl. , 605 (2020) 239–248

work page 2020
[6]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G´ omez Lozano. Decompositions of matrices into diagonaliz- able and square-zero matrices. Linear Multilinear Algebra , 70(19) (2022) 4056–4070

work page 2022
[7]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G. Lozano. Decompositions o f matrices into potent and square-zero matrices. Internat. J. Algebra Comput. , 32(2) (2022) 251–263

work page 2022
[8]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G. Lozano. Decompositions o f matrices into a sum of torsion matrices and matrices of ﬁxed nilpotence. Linear Algebra Appl. , 676 (2023) 44–55. 8 P. DANCHEV, E. GARC ´IA, AND M. G ´OMEZ LOZANO

work page 2023
[9]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G´ omez Lozano. Decompositi ons of matrices into a sum of invertible matrices and matrices of ﬁxed nilpotence. Electron. J. Linear Algebra , 39 (2023) 460–471

work page 2023
[10]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G´ omez Lozano. On prescrib ed characteristic polynomials. arXiv: 2403.15138 [math.RA]

work page arXiv
[11]

Danchev and J

P. Danchev and J. Matczuk. n-Torsion clean rings. Contemp. Math. , 727 (2019) 71–82

work page 2019
[12]

Y. Shitov. The ring M8k+4(Z2) is nil-clean of index four. Indag. Math. (N.S.) , 30 (2019) 1077-1078

work page 2019
[13]

J. ˇSter. On expressing matrices over Z2 as the sum of an idempotent and a nilpotent. Linear Algebra Appl., 544 (2018) 339–349

work page 2018
[14]

J. ˇSter. Nil-clean index of Mn(F2). Linear Algebra Appl. , 632 (2022) 294–307. Institute of Mathematics and Informatics, Bulgarian Acade my of Sciences, 1113 Sofia, Bulgaria Email address : danchev@math.bas.bg Departamento de Matem ´atica Aplicada, Ciencia e Ingenier ´ıa de los Materiales y Tecnolog´ıa Electr ´onica, Universidad Rey Juan Carlos, 28933 M ...

work page 2022

[1] [1]

A. N. Abyzov and D. T. Tapkin. Rings over which every matri ces are sums of idempotent and q-potent matrices. Siberian Math. J. , 62(1) (2021) 1–13

work page 2021

[2] [2]

A. N. Abyzov, D. T. Tapkin. When is every matrix over a ring the sum of two tripotents? Linear Algebra Appl. , 630 (2021) 316–325

work page 2021

[3] [3]

S. Breaz. Matrices over ﬁnite ﬁelds as sums of periodic an d nilpotent elements. Linear Algebra Appl., 555 (2018) 92–97

work page 2018

[4] [4]

Breaz, G

S. Breaz, G. Cˇ alugˇ areanu, P. Danchev and T. Micu. Nil-c lean matrix rings. Linear Algebra Appl., 439 (2013) 3115–3119

work page 2013

[5] [5]

Breaz and S

S. Breaz and S. Megiesan. Nonderogatory matrices as sums of idempotent and nilpotent matrices. Linear Algebra Appl. , 605 (2020) 239–248

work page 2020

[6] [6]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G´ omez Lozano. Decompositions of matrices into diagonaliz- able and square-zero matrices. Linear Multilinear Algebra , 70(19) (2022) 4056–4070

work page 2022

[7] [7]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G. Lozano. Decompositions o f matrices into potent and square-zero matrices. Internat. J. Algebra Comput. , 32(2) (2022) 251–263

work page 2022

[8] [8]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G. Lozano. Decompositions o f matrices into a sum of torsion matrices and matrices of ﬁxed nilpotence. Linear Algebra Appl. , 676 (2023) 44–55. 8 P. DANCHEV, E. GARC ´IA, AND M. G ´OMEZ LOZANO

work page 2023

[9] [9]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G´ omez Lozano. Decompositi ons of matrices into a sum of invertible matrices and matrices of ﬁxed nilpotence. Electron. J. Linear Algebra , 39 (2023) 460–471

work page 2023

[10] [10]

Danchev, E

P. Danchev, E. Garc ´ ıa and M. G´ omez Lozano. On prescrib ed characteristic polynomials. arXiv: 2403.15138 [math.RA]

work page arXiv

[11] [11]

Danchev and J

P. Danchev and J. Matczuk. n-Torsion clean rings. Contemp. Math. , 727 (2019) 71–82

work page 2019

[12] [12]

Y. Shitov. The ring M8k+4(Z2) is nil-clean of index four. Indag. Math. (N.S.) , 30 (2019) 1077-1078

work page 2019

[13] [13]

J. ˇSter. On expressing matrices over Z2 as the sum of an idempotent and a nilpotent. Linear Algebra Appl., 544 (2018) 339–349

work page 2018

[14] [14]

J. ˇSter. Nil-clean index of Mn(F2). Linear Algebra Appl. , 632 (2022) 294–307. Institute of Mathematics and Informatics, Bulgarian Acade my of Sciences, 1113 Sofia, Bulgaria Email address : danchev@math.bas.bg Departamento de Matem ´atica Aplicada, Ciencia e Ingenier ´ıa de los Materiales y Tecnolog´ıa Electr ´onica, Universidad Rey Juan Carlos, 28933 M ...

work page 2022