广义P_0-矩阵及P-矩阵的几个性质

给出了广义线性互补问题中常用到的广义P0矩阵(P矩阵)的几个性质。这些性质类似于通常的半正定矩阵及正定矩阵的性质。矩阵A∈Rn×n为一个半正定(正定)矩阵时,其对角元素是非负(正)的;具有正对角元素的对角矩阵与一个半正定矩阵(正定)的乘积仍为半正定(正定)矩阵;A∈Rn×n为一个P0(P)矩阵的充分必要条件是对任X∈Rn,X≠0,总存在X的某个分量Xi≠0,有Xi(AX)i≥0(>0);若A∈Rn×n是一个半正定矩阵,E为n阶单位矩,则存在某个t>0,使A+tE为一个正定矩阵;而两个半正定(正定)矩阵之和仍为半正定(正定)矩阵。对于类(m1,…,mn)的竖块矩阵N∈Rm0×n,先给出了N的代表子阵的定义,然后得到了广义P0(P)矩阵与它们类似的几个性质。这些性质为更好地解决广义线性互补问题奠定了一定的基础。

Properties of Generalized P0-Matrices and P-Matrices

Some properties of generalized P 0-matrices ( P -matrices) in generalized linear complementarity problems are studied. These properties are similar to properties of positive semi-definite matrices and positive definite matrices. If a matrix A ∈ R n×n is a positive semi-definits matrix(positive matrix), then its diagonal elements are nonnegative (positive); Let D ∈ R n×n be a diagonal matrix, and its diagonal elements are nonnegative, then DA is a positive semi-definite matrix(positive definits matrix). A matrix A ∈ R n×n is a P 0( P ) matrix if and only if there is an index i such that X i ≠0 and X i(AX) i ≥0(>0)for all 0≠ X ∈ R n ; If A ∈ R n×n is a positive semi-definite matrix, E is a unit matrix, then there is a t >0 such that A+tE is a positive definite matrix. The sum of two positive semi-definite (positive definite) matrices is a positive semi-definite (positive definite) matrix. Let N ∈ R m 0×n be a vertical block matrix of type( m 1,…, m n ), a definition of representative submatrix of N is gived, then we propose some properties of generalized P 0-( P ) matrices as same as those of positive semi-definite (positive definite) matrix. The result laid a foundation for solving generalized linear complementarity problems.