Matrix inverse problem and its optimal approximation problem for R-symmetric matrices

Let R is an element of C-nxn be a nontrivial involution, i.e., R-2 = I and R not equal +/- I. A matrix A is an element of C-nxn is called R-symmetric if RAR = A. The solvability conditions and the expression of the matrix inverse problem for R-symmetric matrices with R* = R are derived, also the least-squares solutions of the matrix inverse problem for R-symmetric matrices with R* = R are given. The corresponding optimal approximation problem for R-symmetric matrices with R* = R is considered. We firstly point out that the optimal approximation problem is solvable, then get the expression of its unique solution. It can be seen that this paper generalizes the results mentioned in Zhou [F.-Z. Zhou, L. Zhang, X.-Y. Hu, Least-square solutions for inverse problem of centrosymmetric matrices, Comput. Math. Appl. 45 (2003) 1581-1589]. (c) 2006 Elsevier Inc. All rights reserved.