The Solvability Conditions for the Inverse Problem of Matrices Positive Semidefinite on a Subspace (Journal of Computational Mathematics, 1994)
This paper studies the following two problem:Problem Ⅰ. 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 $S∈R^{n×p}_p$}.Problem Ⅱ. 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 of  Problem Ⅰ.The necessary and sufficient conditions for the solvability of Problem Ⅰ, the expression of the general solution of Problem Ⅰ and the solution of Problem Ⅱ are given for two case. For the general case, the equivalent form of conditions for the solvability of Problem  Ⅰ is given.