The expansion problem of anti-symmetric matrix under a linear constraint and the optimal approximation

This paper mainly discusses the following two problems:Problem I. Given A∈Rn×m,B∈Rm×m,X0∈ASRq×q (the set of q×qq×q anti-symmetric matrices), find X∈ASRn×nX∈ASRn×n such thatATXA=B,X0=X([1:q]),where X([1:q])X([1:q]) is the q×qq×q leading principal submatrix of matrix X.Problem II. Given X*∈Rn×nX*∈Rn×n, find X^∈SE such that∥X*-X^∥=minX∈SE∥X*-X∥,where ∥·∥∥·∥ is the Frobenius norm, and SESE is the solution set of Problem I.The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. Moreover, the optimal approximation solution, an algorithm and a numerical example of Problem II are provided.