矩阵方程AXA^T+BYB^T+AZB^T=D与矩阵方程AXA^T+AZB^T+BZ^TA^T=D的极小范数解(英文)

THE MINIMUM NORM SOLUTIONS OF MATRIX EQUATIONS AXA^T + BYB^T + AZB^T = D AND AXA^T + AZB^T + BZ^TA^T = D

给定A∈Rm×n,B∈Rm×p,D∈Rm×m,设S1={(X,Y,Z)∈SRn×n×SRp×p×Rn×p|AXAT+BYBT+AZBT=D}, S2={(X,Z)∈SRn×n×Rn×p|AXAT+AZBT+BZTAT=D},求(X,Y,Z)∈S1使得‖X‖2+‖Y‖2+‖Z‖2=min及(X,Z)∈S2使得‖2‖2+‖2‖2=min．本文运用矩阵对(A,B)的广义奇异值分解给出了集合S1,S2非空的充分必要条件及X,Y,Z的显式表示．

Given A∈R^m×n,B∈Rm×p,D∈R^m×m,and let S1={（X,Y,Z）∈Sr^n×n×SR^p×p×R^n×p｜AXA^T＋BYB^T＋AZB^T=D},S2={（X,Z）∈SR^n×n×R^n×p｜AXA^T＋AZB^T＋BZ^TA^T=D}.Find （X,Y,Z）∈S1 such that ｜｜X｜｜^2＋｜｜Y｜｜^2＋｜｜Z｜｜^2=min and find （X,Z）∈S2 such that ｜｜X｜｜^2＋｜｜Z｜｜^2=min.By applying the generalized singular value decomposition （GSVD） of the matrix pair （A, B）, the necessary and sufficient conditions under which 81,82 are nonempty are given. The explicit expressions of X,Y,Z are presented.