关于矩阵的积和式

来源期刊：中南大学学报(自然科学版)2003年第6期

论文作者：亢保元

文章页码：711 - 713

关键词：矩阵;积和式;行列式

Key words：matrices; permanents; determinants

摘    要：基于对方阵积和式性质的讨论和积和式概念的推广,运用极限的思想给出了一个逐步降阶而计算积和式的思路.通过引入复杂积的概念,给出了积和式与行列式之间的关系.得出:若A为n阶方阵,P和Q均为n阶对角阵,则Per(PAQ)=Per(P)·Per(A)·Per(Q);若n阶方阵A有形式[1αα^TB],其中α=(1,…,1)为n-1维行向量,则PerA=PerB+σ^n-2(B);若A为方阵,则(PerA)²=|A|²+4ComA.

Abstract: Some properties for the permanents of matrices were given, the definition of the permanents were generalized,the method of the computation for the permanents of matrices was studied, and the relation between the permanents and determinants of square matrices was studied. The results are shown as follows: Suppose A is a n×n square matrices,P and Q are n×n diagonal square matrices, then Per(PAQ)=Per(P)·Per(A)·Per(Q); Suppose n×n square matrices A=[1αα^TB],α=(1,…,1) isn-1 dimension rowvector, then PerA=PerB+σ^n-2(B); then Suppose A is a square matrices, then (PerA)²=|A|²+4ComA, ComA is complex product of A.