广义Z-矩阵及M-矩阵的几个性质

Properties of Generalized Z - Matrices and M - Matrices

给出了广义线性互补问题中常用到的广义Z-矩阵及M-矩阵的几个性质。这些性质类似于通常意义下的Z-矩阵及M-矩阵的性质。矩阵A∈R^(n×n)为一个Z-矩阵的充分必要条件是对于某矩阵P∈R^(n×n),P≥0,以及某实数a∈R,使得A=aE-P;A∈R^(n×n)为一个M-矩阵当且仅当A同时为Z-矩阵和P-矩阵;若A是一个Z-矩阵,A是一个具有正对角元的对角矩阵,则M=AA仍是一个Z-矩阵。两个Z-矩阵的和是一个Z-矩阵。对于类(m_1,…,m_n)的竖块矩阵N∈R^(m_0×n),先给出了N的代表子阵的定义,然后得到了广义Z-矩阵及M-矩阵与它们类似的几个性质及其几个等价性结论。这为更好的解广义线性互补问题奠定了一定的基础。

Some properties of generalized Z - matrices and M - matrices in generalized linear complementarity problems were studied. These properties are similar to properties of square 2 - matrices and M - matrices. A matrix A ∈ Rn×n is a Z - matrix if and only if there is a matrix P ∈ Rn×n, P ≥0 , and a α 6 R such that A = αE - P ; a matrix A α Rn×n is a M - matrix if and only if matrix A is a Z - matrix and P - matrix ; Let A ∈ Rn×n be a Z - matrix and A is a diagonal matrix, and its diagonal elements are positive, then M = AA is a Z - matrix; The sum of two Z - matrices is a Z - matrix. Let N∈ Rn0×n be a vertical block matrix of type (m1,…, mn), a definition of representative submatrix of N was given, then we propose some properties and theorems of generalized Z - matrices and M - matrices as same as those square Z - matrices and M - matrices. The result laid a foundation for solving generalized linear complementarity problems.