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Kijti Rodtes

Publications and source records attributed to Kijti Rodtes.

18 recordsLinked to original sources
arXiv2023

Idempotent factorization on some matrices over quadratic integer rings

In 2020, Cossu and Zanardo raised a conjecture on the idempotent factorization on singular matrices in the form $\begin{pmatrix} p&z\\ \bar{z}&\sfrac{\lVert z\rVert}{p} \end{pmatrix},$ where $p$ is a prime integer which is irreducible but not prime element in the ring of integers $\mathbb{Z}[\sqrt{D}]$ and $z\in\mathbb{Z}[\sqrt{D}]$ such that $\langle p,z\rangle$ is a non-principal ideal. In this paper, we provide some classes of matrices that affirm the conjecture and some classes of matrices that oppose the conjecture. We further show that there are matrices in the above form that can not be written as a product of two idempotent matrices.

math.RA↗
arXiv2023

Some remarks on permanent dominant conjecture

In this paper we provide an identity between determinant and generalized matrix function. Also, a criterion of positive semi-definite matrices affirming the permanent dominant conjecture is given. As a consequence, infinitely many infinite classes of positive semi-definite serving the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column.

math.RA↗
arXiv2022

Chollet's permanent conjecture for $4\times 4$ matrices

In the year 1982, John Chollet conjectured that, for any pair of $n\times n$ positive semidefinite matrices $A,B$, $per(A)\cdot per(B)\geq per(A\circ B)$, where $A\circ B$ is the Hardamard product of $A$ and $B$. This conjecture was proved to be valid for $n=2, 3$ in the year 1987. In this paper, we show that the conjecture holds true for $n=4$.

math.RA↗
arXiv2022

Matrices over commutative rings as sum of higher powers

On the Waring's problems for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as a sum of $k$-th powers with $k=2,3,4,5,6,7,8$ in several literatures. In this paper, we provide the similar conditions for matrices written as a sum of $k$-th powers with $k=9,10,11,12,13,14,15,16$.

math.RA↗
arXiv2021

Determinants preserving maps on the spaces of symmetric matrices and skew-symmetric matrices

Denote $Σ_n$ and $Q_n$ the set of all $n \times n$ symmetric and skew-symmetric matrices over a field $\mathbb{F}$, respectively, where $char(\mathbb{F})\neq 2$ and $\lvert \mathbb{F} \rvert \geq n^2+1$. A characterization of $ϕ,ψ:Σ_n \rightarrow Σ_n$, for which at least one of them is surjective, satisfying $$\det(ϕ(x)+ψ(y))=\det(x+y)\qquad(x,y\in Σ_n)$$ is given. Furthermore, if $n$ is even and $ϕ,ψ:Q_n \rightarrow Q_n$, for which $ψ$ is surjective and $ψ(0)=0$, satisfy $$\det(ϕ(x)+ψ(y))=\det(x+y)\qquad(x,y\in Q_n),$$ then $ϕ=ψ$ and $ψ$ must be a bijective linear map preserving the determinant.

math.RA↗
arXiv2020

Divisibility of LCM matrices by Totally nonnegative GCD matrices

In this paper, we show that all totally nonnegative GCD matrices are always divisors of the corresponding LCM matrices in the ring $\mathbb{M}_{n}(\mathbb{Z})$. We also introduce \lq\lq column monotone matrices" used to construct all totally nonegative GCD matrices.

math.RA↗
arXiv2019

Generalized matrix functions on a linear sum of permutation matrices and their cousins

A generalized matrix function is a generalization of determinant and permanent function. In this paper, we introduced the formula for the value of a generalized matrix function of a linear sum of permutation matrices. We show that a linear sum of permutation matrices satisfies the permanent dominance conjecture. Finally, we apply the result to some cousins of permutation matrices.

math.RA↗
arXiv2018

On the character tables of symmetric groups

In this paper, some zeros and non-zeros in the character tables of symmetric groups are displayed in the partition forms. In particular, more zeros of self conjugate partitions beside odd permutations are heavily investigated.

math.RT↗
arXiv2018

The equality of generalized matrix functions on the set of all symmetric matrices

A generalized matrix function $d_χ^G : M_n(\mathbb{C}) \rightarrow \mathbb{C}$ is a function constructed by a subgroup $G$ of $S_n$ and a complex valued function $χ$ of $G$. The main purpose of this paper is to find a necessary and sufficient condition for the equality of two generalized matrix functions on the set of all symmetric matrices, $\mathbb{S}_n(\mathbb{C})$. In order to fulfill the purpose, a symmetric matrix $S_σ$ is constructed and $d_χ^G(S_σ)$ is evaluated for each $σ\in S_n$. By applying the value of $d_χ^G(S_σ)$, it is shown that $d_χ^G(AB) = d_χ^G(A)d_χ^G(B)$ for each $A, B \in \mathbb{S}_n(\mathbb{C})$ if and only if $d_χ^G = \det$. Furthermore, a criterion when $d_χ^G(AB) = d_χ^G(BA)$ for every $A, B \in \mathbb{S}_n(\mathbb{C})$, is established.

math.RA↗
arXiv2017

Inverse Eigenvalue Problem of Cell Matrices

In this paper, we consider the problem of reconstructing an $n \times n$ cell matrix $D(\vec{x})$ constructed from a vector $\vec{x} = (x_{1}, x_{2},\dots, x_{n})$ of positive real numbers, from a given set of spectral data. In addition, we show that the spectrum of cell matrices $D(\vec{x})$ and $D(π(\vec{x}))$ are the same, for every permutation $π\in S_{n}$.

math.RA↗
arXiv2016

Symmetry classes of tensors and Semi-direct product of finite abelian groups

In the study of symmetry classes of tensors, finding examples of symmetry classes of tensors that possess an o*-basis is of considerable interest. There are only few classes of groups that have been provided a necessary and sufficient condition for having such a basis. There is no general criterion for any finite groups yet. In this note, we provide a necessary and sufficient condition for the existence of o*-basis of symmetry classes of tensors associated with semi-direct product of some finite abelian groups and, consequently, their wreath product.

math.RT↗
arXiv2015

Orthogonal bases of Brauer symmetry classes of tensors for groups having cyclic support on non-linear Brauer characters

This paper provides some properties of Brauer symmetry classes of tensors. We derive a dimension formula for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups having cyclic groups support on non-linear Brauer characters. Using the derived formula, we investigate the necessary and sufficient condition for the existence of the o-basis of Dicyclic groups, Semi-dihedral groups and also reinvestigate those things on Dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors associated to those groups are also included.

math.GR↗
arXiv2015

Orthogonal bases of Brauer relative symmetric polynomials for certain groups

In this paper, we discuss O-basis of symmetry classes of polynomials associated with the Brauer character of the Semi-Dihedral groups and Dihedral groups. Also, necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the Semi-Dihedral groups.

math.CV↗