中文摘要：

本文给出了矩阵反问题AX=B具有可对称正定化解与可对称半正定化解的必要充分条件，得到了通解的表达式，同时解决了方程的对称半正定化解对己给矩阵的最佳逼近问题。

英文摘要：

In this paper, the necessary and sufficient conditions for the existence of and the expresssions for the positive definite （semidefinite） symmetrizable solutions ofinverse problems AX = B are obtained, and the optimal approximation semidefinite symmetrizable solution to given marices is also derived.