Inverse problems for symmetric matrices with a submatrix constraint Original Research Article

This paper is concerned with the following problems: Problem I(a). Given a full column rank matrix X∈Rn×pX∈Rn×p and symmetric matrices B∈Rp×pB∈Rp×p and A0∈Rr×rA0∈Rr×r, find an n×nn×n symmetric matrix A such that View the MathML sourceXTAX=B,A([1,r])=A0, Turn MathJax on where A([1,r])A([1,r]) is the r×rr×r leading principal submatrix of the matrix A. Problem I(b). Given a matrix X∈Rn×pX∈Rn×p and symmetric matrices B∈Rp×pB∈Rp×p, A0∈Rr×rA0∈Rr×r, find an n×nn×n symmetric matrix A such that View the MathML source‖XTAX−B‖=min,s.t. A([1,r])=A0. Turn MathJax on Problem II. Given an n×nn×n symmetric matrix View the MathML sourceA˜, find View the MathML sourceAˆ∈SE such that View the MathML source‖A˜−Aˆ‖=infA∈SE‖A˜−A‖, Turn MathJax on where SESE is the solution set of Problem I(a). By applying the generalized singular value decomposition (GSVD) and the canonical correlation decomposition (CCD) of a matrix pair, the solvability conditions for Problem I(a) and the general forms of the solution of Problem I are presented. The expression of the solution of Problem II is derived. A numerical algorithm for solving Problem II is provided.