arXiv subjects

Ratsiri Sanguanwong

Publications and source records attributed to Ratsiri Sanguanwong.

4 recordsLinked to original sources

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

A remark on an equivalence of two versions of Polya's permanent problem

In 2004, some equivalent versions of Polya's permanent problem were listed in 24 versions. However, there is a flaw on the theorem that affirms an equivalence of version 11 and 12. In order to correct the slip, we provide a characterization when the determinant and permanent of a given nonnegative matrix are equal. Moreover, the new characterization yields necessary and sufficient conditions for the equality of the determinant and permanent of powers of a given nonnegative matrix.

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

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↗