四元数体上矩阵方程{AXA^＊=B CXC^＊=D的非负定解

Nonnegative definite solution of the matrix equation AXA*＝B CXC*＝D over quaternion field

本文讨论四元数体上矩阵方程AXA*=BCXC*=D的非负定解,解决了以下问题:(1)给出了矩阵方程AXA*=BCXC*=D存在非负定解的充分必要条件;(2)当矩阵方程AXA*=BCXC*=D有非负定解时,给出了通解的表达式;(3)当矩阵方程AXA*=BCXC*=D有非负定解X时,给出了X的秩的范围.

This paper discuses the nonnegativedefinite solution of the matrix equation \$AXA=B CXC=D\$over quaternion field. The following problems are solved: (1) The necessary and sufficient condition for the existence of the nonnegativedefinite solution to the matrix equation\$AXA=B CXC=D\$;(2) If the equation\$AXA=B CXC=D\$is consistent, it gives a representation of the general solution to\$AXA=B CXC=D\$;(3) If X is an nonnegativedefinite solution to\$AXA=B CXC=D\$,it gives the formula for the rank of X.