五对角线逆M-矩阵的Hadamard积(英文)

Hadamard Product for Five-diagonal Inverse M-matrices

令M-1记所有n×n逆M矩阵的集合,Sk(k>1)记所有实矩阵其每个k×k主子矩阵都是逆M矩阵的集合.首先证得如果A,B∈M-1分别是上、下Hessenberg矩阵,则对任意H1,H2∈S2,AB和(AH1)(BH2)都是三对角线矩阵(因而是完全非负矩阵);其次证得如果A=(aij),B=(bij)(M-1满足aji=bij=0,i-j≥3,则对任意H1,H2∈S3,AB和(AH1)(BH2)都是五对角线逆M矩阵.

Let M^-1  be the set of all n×n inverse M-matrices;S_k be the set of all n×n real matrices A such that each k×k principal submatrix of A is in M^-1 . Firstly we show that:if A,B∈M^-1  are lower and upper Hessenberg matrices,respectively,then AB and (AH_1)(BH_2) are tridiagonal inverse M-matrices which are totally nonnegative for any H_1,H_2∈S_2. Secondly we show that:if A=(a_~ij ),B=(b_~ij )∈M^-1  satisfy a_~ji =b_~ij =0,i-j≥3, then AB and (AH_1)(BH_2) are five-diagonal inverse M-matrices for any H_1,H_2∈S_3.