The solvability conditions for the inverse problems of symmetric ortho-symmetric matrices

摘要

This paper studies the following two problems:Problem IGiven X, B∈Rn×m, find A∈SRnP such that AX=B, whereSRnP={A∈SRn×n|PA∈SRn×n,forgivenP∈ORn×nsatisfyingPT=P}. Problem IIGiven Ã∈Rn×n, find A∗∈SE such that∥Ã−A∗∥=infA∈SE∥Ã−A∥,where ∥·∥ is the Frobenius norm, and SE is the solution set of Problem I.Necessary and sufficient conditions for the solvability of Problem I and the general form of the solution of Problem I are given. For Problem II, the expression of the solution is provided. Furthermore, it is pointed that some results of [Math. Numer. Sin. 1 (2000) 29] are special cases of this paper.