Invertible matrices under non-Archimedean absolute value and its inverse

Abstract

Invertible matrices play a crucial role in various areas of mathematics, science and engineering. Although there are many ways to determine whether a matrix is invertible or not, it is still a fundamental and an important research work in linear algebra, especially in large-scale numerical computations. Based on the non-Archimedean absolute value, we in this paper present a new class of invertible matrices called doubly strictly diagonally dominant matrices under non-Archimedean absolute value (DSDD${}_{n.A.}$ matrices). It includes the strictly diagonally dominant matrices under non-Archimedean absolute value (SDD${}_{n.A.}$ matrices) presented by Nica and Sprague in [The American Mathematical Monthly, 130 (2023) 267-275]. Some examples are given to show the relationships of SDD${}_{n.A.}$ matrices, DSDD${}_{n.A.}$ matrices, SDD matrices (strictly diagonally dominant matrices under Archimedean absolute value), DSDD matrices (doubly strictly diagonally dominant matrices under Archimedean absolute value), and H-matrices. Moreover, it is proved that the inverse of DSDD${}_{n.A.}$ matrices is also DSDD${}_{n.A.}$ in some cases, which displays remarkable difference with the inverse of DSDD matrices and SDD matrices.