Products of Hermitian invertible matrices over local commutative rings

Hau, Nguyen Thi Truc; Ngoc, Nguyen Kim

doi:10.22060/ajmc.2026.25008.1493

Products of Hermitian invertible matrices over local commutative rings

Articles in Press

Document Type : Original Article

Authors

Nguyen Thi Truc Hau

Nguyen Kim Ngoc

Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

10.22060/ajmc.2026.25008.1493

Abstract

Let $R$ be a local commutative ring with an involution $*$ and let $n$ be an integer greater than $1$. A matrix $(a_{ij})\in \mathrm{M}_n(R)$ is called Hermitian if $a_{ij}^*=a_{ji}$ for all $1\le i, j \le n$. One of the main results of the paper is to prove that if the residue field $\overline{R}$ of $R$ is infinite, then a matrix $A$ in the general linear group $\mathrm{GL}_n(R)$ is a product of five Hermitian matrices if and only if $\mathrm{det}(A)^*=\mathrm{det}(A)$.

Keywords

Hermitian matrices

matrix factorization

local commutative ring

Subjects