The least squares problem of the matrix equation A_1X_1B_1~T+A_2X_2B_2~T=T  

作　　者：QIU Yu-yang ZHANG Zhen-yue WANG An-ding 

机构地区：[1]Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China [2]College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China [3]College of Information and Electronics, Zhejiang Gongshang University, Hangzhou 310018, China

出　　处：《Applied Mathematics(A Journal of Chinese Universities)》2009年第4期451-461,共11页高校应用数学学报（英文版）（B辑）

基　　金：supported in part by the Social Science Foundation of Ministry of Education(07JJD790154);the National Science Foundation for Young Scholars (60803076);the Natural Science Foundation of Zhejiang Province (Y6090211);Foundation of Education Department of Zhejiang Province (20070590);the Young Talent Foundation of Zhejiang Gongshang University

摘　　要：The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.

关 键 词：least squares problem generalized inverse solution set general solutions parameter matrix 

分 类 号：O151.21[理学—数学] O241.5[理学—基础数学]