矩阵方程AXA^T+BYB^T=C的对称最优解及最佳逼近

THE OPTIMAL SYMMETRIC SOLUTION AND THE BEST APPROXIMATION OF MATRIX EQUATION AXA^T+BYB^T=C

考虑以下问题:问题1:给定A∈Rm×n,B∈Rm×l,C∈Rm×m,L={(X,Y)|AXAT+BYBT=C,X∈SRn×n,Y∈SRl×l}≠φ,找(⌒X,⌒Y)∈L,使得‖(⌒X,⌒Y)‖=(‖X‖2+‖Y‖2)12=min.问题2:任意给定X∧∈Rn×n,Y∧∈Rl×l,找(X∧,Y∧)∈L,使得‖X∧-~X‖2+‖Y∧-~Y‖2=min(X,Y)∈L(‖X-~X‖2+‖Y-Y~‖2).讨论了矩阵方程AXAT+BYBT=C有解的充要条件,得到了L的具体表达式,给出了问题1与问题2的唯一解证明与显式表示.

In this paper, the following problems are considered：Problem Ⅰ Given matrix A∈R^m×n,B∈R^m×l,C∈R^m×m,L={（X,Y）｜AXA^T＋BYB^T=C,X∈SR^n×n,Y∈SR^l×l}≠φ,find（X⌒,Y⌒）∈L,such that‖（X⌒,Y⌒）‖=（‖X‖^2＋‖Y‖^2）（1）/（2）=min.problemⅡ：Given matrix X^∈R^n×n,Y^∈R^l×l,找（X^,Y^）∈L,such that ‖X^-X^～‖^2＋‖Y^-Y^～‖^2=min（X,Y）∈L（‖X-X^～‖^2＋‖Y-Y^～‖^2）. A necessary and sufficient condition for matrix equation AXA^T ＋BYB^T= C having symmetric solution is discussed and the expressions of L are obtained. The unique solution existing in problems Ⅰ and Ⅱ are proved and the explicit representation of solution is given.