The inverse problem of nonsymmetric matrices with a submatrix constraint and its approximation

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): Given matrices X ∈ Rn×p,B ∈ Rp×p and A0 ∈ Rr×r, find a matrix A ∈ Rn×n such that {norm of matrix}XTAX - B{norm of matrix} = min, s. t. A([1, r]) = A0, where A([1, r]) is the r×r leading principal submatrix of the matrix A. We then consider a best approximation problem: given an n × n matrix à with Ã([1, r]) = A0, find  ∈ SE such that {norm of matrix} à - Â{norm of matrix} = minA∈SE {norm of matrix} à - A{norm of matrix}, where SE is the solution set of LSP. We show that the best approximation solution  is unique and derive an explicit formula for it.