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Esther García

Publications and source records attributed to Esther García.

8 recordsLinked to original sources
arXiv2026

Leibniz algebras and their connection to Jordan pair disystems

In this paper we present a generalization of Jordan pairs, called Jordan pair disystem, which appear naturally as the wings of a Leibniz algebra with a finite $\mathbb{Z}$-grading; conversely, we give the Tits-Kantor-Koecher construction for Jordan pair disystems, obtaining a Leibniz algebra with a short $\mathbb{Z}$-grading. This construction extends the construction given by Gubarev and Kolesnikov for Jordan dialgebras (arXiv:0907.1740). We introduce homotopes for Jordan pair disystems at elements and we show that we obtain again a Jordan dialgebra. Moreover, we introduce abelian inner ideals and their kernels for Leibniz algebras, and prove that the subquotient of a Leibniz algebra with respect to an abelian inner ideal is a Jordan pair disystem. This notion of subquotient extends the construction of Jordan dialgebras at $Q$-Jordan elements given by Felipe and Velásquez.

math.RA↗
arXiv2026

Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices

We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. Our main theorem of that type, combined with results from our recent publication in Linear Algebra & Appl. (2026) (see [7]), totally settle this problem for all finite fields different from $\mathbb{F}_2$ and $\mathbb{F}_3$. However, in this paper we also prove that each matrix over $\mathbb{F}_2$ is expressible as the sum of a potent matrix with index of potency not exceeding 4 and a nilpotent matrix with index of nilpotency not exceeding 2, thus substantiating recent examples due to Šter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]).

math.RA↗
arXiv2025

Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices

Let $\mathbb{F}$ be a finite field of odd characteristic. When $|\mathbb{F}|\ge 5$, we prove that every matrix $A$ admits a decomposition into $D+M$ where $D$ is diagonalizable and $M^2=0$. For $\mathbb{F}=\mathbb{F}_3$, we show that such decomposition is possible for non-derogatory matrices of order at least 5, and more generally, for matrices whose first invariant factor is not a non-zero trace irreducible polynomial of degree 3; we also establish that matrices consisting of direct sums of companion matrices, all of them associated to the same irreducible polynomial of non-zero trace and degree 3 over $\mathbb{F}_3$, never admit such decomposition. These results completely settle the question posed by Breaz in Lin. Algebra & Appl. (2018) asking if it is true that for big enough positive integers $n\ge 3$ all matrices $A$ over a field of odd cardinality $q$ admit decompositions of the form $E+M$ with $E^q=D$ and $M^2=0$: the answer is {\it yes} for $q\ge 5$, but there are counterexamples for $q=3$ and each order $n=3k$, $k\ge 1$.

math.RA↗
arXiv2024

Decompositions of Periodic Matrices into a Sum of Special Matrices

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.

math.RA↗
arXiv2024

On prescribed characteristic polynomials

Let $\mathbb{F}$ be a field. We show that given any $n$th degree monic polynomial $q(x)\in \mathbb{F}[x]$ and any matrix $A\in\mathbb{M}_n(\mathbb{F})$ whose trace coincides with the trace of $q(x)$ and consisting in its main diagonal of $k$ 0-blocks of order one, with $k<n-k$, and an invertible non-derogatory block of order $n-k$, we can construct a square-zero matrix $N$ such that the characteristic polynomial of $A+N$ is exactly $q(x)$. We also show that the restriction $k<n-k$ is necessary in the sense that, when the equality $k=n-k$ holds, not every characteristic polynomial having the same trace as $A$ can be obtained by adding a square-zero matrix. Finally, we apply our main result to decompose matrices into the sum of a square-zero matrix and some other matrix which is either diagonalizable, invertible, potent or torsion.

math.RA↗
arXiv2023

Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence

For $n\ge 2$ and fixed $k\ge 1$, we study when a square matrix $A$ over an arbitrary field $\mathbb{F}$ can be decomposed as $T+N$ where $T$ is a torsion matrix and $N$ is a nilpotent matrix with $N^k=0$. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of $A\in \mathbb{M}_{n}(\mathbb{F})$ is algebraic over its base field and the rank of $A$ is at least $\frac nk$, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for $k=2$ and nilpotent matrices over arbitrary fields (even over division rings). 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).

math.RA↗
arXiv2014

A Perron-Frobenius theory for block matrices associated to a multiplex network

The uniqueness of the Perron vector of a nonnegative block matrix associated to a multiplex network is discussed. The conclusions come from the relationships between the irreducibility of some nonnegative block matrix associated to a multiplex network and the irreducibility of the corresponding matrices to each layer as well as the irreducibility of the adjacency matrix of the projection network. In addition the computation of that Perron vector in terms of the Perron vectors of the blocks is also addressed. Finally we present the precise relations that allow to express the Perron eigenvector of the multiplex network in terms of the Perron eigenvectors of its layers.

physics.soc-ph↗