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 ‖XTAX − B‖ = 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 ‖à − Â‖ = minA∈SE ‖Ã− A‖, where SE is the solution set of LSP. We show that the best approximation solution  is unique and derive an explicit formula for it. Keywords—Inverse problem, Least-squares solution, model updating, Singular value decomposition(SVD), Optimal approximation.