子空间上矩阵方程AXB=D的对称解

Symmetric Solutions of Linear Matrix Equation AXB=D in Subspace

设Ω={z∈RnGz=o,G∈Rk×n},SRΩn×n={X∈Rn×nzT(X-XT)z=0,z∈Ω}。本文给出了矩阵方程AXB=D有解X∈SRnΩ×n的充分必要条件,并在有解的情况下,给出了通解的显式表示。

LetΩ={z∈R^n｜Gz=0,G∈R^k×n},SRΩ^n×n={x∈R^n×n｜z^T（x-x^T）z=0,Vz∈Ω}, the sufficient and necessary conditions for the linear matrix equation AXB=D with a solution X∈SRΩ^n×n are derived. The explicit general solutions are given when the conditions are met.