一类广义中心对称矩阵的Procrustes问题

Procrustes Problems for a Class of Generalized Centrosymmetric Matrices

设R∈n×n为广义反射矩阵满足R=RH=R-1≠±In.若G∈n×n满足RGR=G,则称G为广义中心对称矩阵。所有n×n阶广义中心对称矩阵的全体记为GCS n×n。考虑问题Ⅰ:给定X,Y,D∈n×p,求A,B∈GCS n×n,使得‖AX-BY-D‖=min。问题Ⅱ:给定A,B∈n×n,求(^A,^B)∈φ(X,Y,D)使得‖(^A,^B)-(A,B)‖=min(A,B)∈φ(X,Y,D)‖(A,B)-(A,B)‖(φ(X,Y,D)是问题Ⅰ的解集合)。文中给出了问题Ⅰ的通解表示及问题Ⅱ的唯一解^A,^B的表达式。

Let R∈^(n×n) be a nontrivial generalized reflection matrix satisfying R=R^H=R^(-1)≠±I_n.G∈^(n×n) is said to be the generalized centrosymmetric if RGR=G.The set of all n×n generalized centrosymmetric matrices is denoted by GCS^(n×n).GivenX,Y,D∈^(n×p),the matrices A,B∈GCS(^(n×n)) that can minimize ‖AX-BY-D‖(Frobenius norm),are characterized.Givne arbitrary ,∈^(n×n),the unique matrix (,) among the minimizers of ‖(AX-BY-D‖ in GCS^(n×n) that can minimize ‖(,)-(A,B)‖ are found.