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 A ∈ R n × m , B ∈ R m × m , X 0 ∈ ASR q × q (the set of q × q anti-symmetric matrices), find X ∈ ASR n × n such that A T XA = B , X 0 = X ( [ 1 : q ] ) , where X ( [ 1 : q ] ) is the q × q leading principal submatrix of matrix X. m II X * ∈ R n × n , find X ^ ∈ S E such that ∥ X * - X ^ ∥ = min X ∈ S E ∥ X * - X ∥ , where ∥ · ∥ is the Frobenius norm, and S E is the solution set of Problem I. cessary 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.