子空间上部分对称半正定和亚半正定矩阵反问题有解的条件

Solvability Conditions for Inverse Problem of Part Symmetric Positive Semidefinite and Part Semipositive Subdefinite Matrices on Subspace

本文考虑以下问题:问题Ⅰ:给定G∈Rn×p,X,B∈Rn×m,求A∈GSRn≥×0n使得AX=B,其中:GSRn≥×0n={A∈Rn×n|xTAx≥0且xT(A-AT)=0,x∈R(G)}。问题Ⅱ:给定G∈Rn×p,X,B∈Rn×m,求A∈GRn≥×0n使得AX=B,其中GRn≥×0n={A∈Rn×n|xTAx≥0,x∈R(G)}。讨论了问题Ⅰ与问题Ⅱ有解的充要条件,并在有解时给出了通解的一般表达式。

Two problems are considered as follows. Problem Ⅰ： Given G∈R^n×p,X,B∈R^nm,find A∈GSR^n×n≥0,such that AX=B,Where：GSR^n×n≥0={A∈R^n×n｜x^TAx≥0且x^T（A-A^T）=0,arbitary d x∈R（G）}.Problem Ⅱ：Given G∈R^n×P,X,B∈R^n×n,find A∈GR^n×n ≥0such that AX=B,where GRn×n≥0={A∈Rn×n｜x^TAx≥0,arbitary x∈R（G）} The necessary and sufficient conditions of the problem Ⅰ and problem Ⅱ having a solution are discussed. The expressions for the general solution of the problem Ⅰ and problem Ⅱ are given.