The nearness problems for symmetric matrix with a submatrix constraint

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): given a matrix X ∈ R n × p and symmetric matrices B ∈ R p × p , A 0 ∈ R r × r , find an n × n symmetric matrix A such that ∥ X T AX - B ∥ = min , s.t. A ( [ 1 , r ] ) = A 0 , 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 symmetric matrix A ˜ with A ˜ ( [ 1 , r ] ) = A 0 , find A ^ ∈ S E such that ∥ A ˜ - A ^ ∥ = min A ∈ S E ∥ A ˜ - A ∥ , where S E is the solution set of LSP. We show that the best approximation solution A ^ is unique and derive an explicit formula for it.