The Study Of Some Special Matrices Over Commutative Semirings

Matrix and determinant are essential concepts which is used to solve linear equations.This paper mainly considers the application of generalized orthogonal matrices and determinant of matrices over a commutative semiring.We first discuss some properties of generalized orthogonal matrices and their equivalent characterization under some additional conditions.And second,we obtain some necessary and sufficient conditions for which a standard orthogonal set can be extended to a standard orthogonal basis in a finitely generated semimodule over a commutative semiring.Then,the properties of determinants of matrices are given,and some necessary and sufficient conditions that a system of linear equation is solvable with certain conditions are studied.At last,we show that the Cramer's rule over a commutative semiring is valid.This paper is organized as follows:In Chapter 1,some basic concepts and related conclusions of semirings are reviewed.In Chapter 2,we first study the properties of generalized orthogonal matrices over a commutative semiring and their equivalent characterization under some additional conditions,then some necessary and sufficient conditions that a standard orthogonal set can be extended to a standard orthogonal basis in a finitely generated semimodule over a commutative semiring are obtained.In Chapter 3,we investigate some properties of the determinant of a matrix over a commutative semiring and give some necessary and sufficient conditions that a system of linear equation is solvable with certain conditions.It finally shows that the Cramer's rule is valid.