The Solvability Conditions for the Inverse Problem of Matrices Positive Semidefinite on a Subspace

This paper studies the following two problem:ProblemⅠ. Given $X,B∈R^{n×m}$, find $A∈P_{s,n}$, such that $AX=B$, where $P_{s,n}$={$A∈SR^{n×n}|x^T AX≥0，?S^Tx=0$ , for given $S∈R^{n×p}_p$}.ProblemⅡ. Given $A^*∈R^{n×n}$, find $\hat{A}∈S_E$, such that $||A^*-\hat{A}||$=inf$_{A∈S_E}||A^*-A||$ where $S_E$ denotes the solutions set ofProblemⅠ.The necessary and sufficient conditions for the solvability ofProblemⅠ, the expression of the general solution ofProblemⅠ and the solution ofProblemⅡ are given for two case. For the general case, the equivalent form of conditions for the solvability ofProblemⅠ is given.