半正定分块矩阵和一个线性矩阵方程及其反问题

Positive Semidefinite Partitioned Matrices and a LinearMatrix Eqnation and its Inverse Problem

中文摘要:

      一个实的（未必对称）ｎ×ｎ矩阵Ａ称为半正定的，如果对任意非零的ｎ维行向量ｘ，均有ｘＭｘ^t≥０．本文给出了一个分块ｎ×ｎ矩阵为半正定的充要条件．另外，我们讨论了线性矩阵方程ＡＸ＝Ｂ对解附加种种条件下的解．我们应用矩阵在相抵下的标准形给出了这一方程的相容性的充要条件．还给出这个方程的反问题在对解附加各种条件下的解．

英文摘要:

      A real (may not symmetric) n×n matrix M is said to be positive semidefiniteif, for any real nonzero n diniensional row vector x, xMx^t≥0. In this paper, we givea necessary and sufficient condition for determining whether a partitioned n ×n matrixis positive semidefinite. Moreover, we consider the solutions of linear matrix equationAX = B with variant conditions on the solutions. The necessary and sufficient conditionsfor the consistency of this equation are derived using the canonical form of a matrix under the equivalence. The iverse problem of this equation with variant couditions on the solutions is also included.