Mathematica Eterna
Open Access

Abstract

The matrix in semilinear spaces over commutative semirings

Wenxin Luo, Chunchan Weng

In this paper, some properties of matrices over commutative semirings are researched deeply. We extend the theorem about invertible matrix and show a necessary condition that a matrix is invertible.And we discuss in n-dimensional L-semilinear space Vn every vector of Vn can be uniquely represented by a linear combination of any basis of Vn. On the other hand, we show the connection between two bases of Vn with the transition matrix and prove an inequality in case that the rank of the matrix is redefined over commutative semirings. We give the proof that a set of linearly independent vectors is still linearly independent under semilinear transformation. We prove that some theorems of the determinant of a matrix still exist for the permanent, but some of the theorems do not. We show the necessary and sufficient condition that the permanent of an invertible matrix is zero.