On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

Danchev, Peter V.

doi:10.17516/1997-1397-2021-14-5-547-553

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…Further insight in that matter over some special finite rings was achieved by us in [9]. We also refer the interested reader to [8] for some other aspects of the realization of matrices into the sum of specific elements over certain fields.…”

Section: Introduction and Fundamentalsmentioning

confidence: 99%

Decomposition of matrices into invertible and square-zero matrices

Danchev¹,

Garcı́a²,

Lozano³

2023

Preprint

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In this paper we show that any matrix A in Mn(F) over an arbitrary field F can be decomposed as a sum of an invertible matrix and a nilpotent matrix of order at most two if and only if its rank is at least n 2 . We also study when A can be decomposed as the sum of a torsion matrix and a nilpotent matrix of order at most two. For fields of prime characteristic, we show that this second decomposition holds as soon as the characteristic polynomial of A is algebraic over its base field and the rank condition is fulfilled, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve the second decomposition problem for nilpotent matrices over arbitrary fields. This somewhat continues our recent publications in Lin. & Multilin. Algebra (2023) and Internat. J. Algebra & Computat. ( 2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra & Appl. (2016).

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Section: Introduction and Fundamentalsmentioning

confidence: 99%

Decomposition of matrices into invertible and square-zero matrices

Danchev¹,

Garcı́a²,

Lozano³

2023

Preprint

View full text Add to dashboard Cite

show abstract

Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence

Danchev

Garcı́a

Lozano

2023

Linear Algebra and its Applications

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Decompositions of matrices into potent and square-zero matrices

Danchev

Garcı́a

Lozano

2022

Int. J. Algebra Comput.

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In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Linear Multilinear Algebra (2022) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), Šter in Linear Algebra Appl. (2018) and Shitov in Indag. Math. (2019).

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On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

Cited by 3 publications

References 12 publications

Decomposition of matrices into invertible and square-zero matrices

Decomposition of matrices into invertible and square-zero matrices

Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence

Decompositions of matrices into potent and square-zero matrices

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