Inverse problems for symmetric matrices with a submatrix constraint

This paper is concerned with the following problems: Problem I(a). Given a full column rank matrix [email protected]?R^n^x^p and symmetric matrices [email protected]?R^p^x^p and A"[email protected]?R^r^x^r, find an nxn symmetric matrix A such thatX^TAX=B,A([1,r])=A"0, where A([1,r]) is the rxr leading principal submatrix of the matrix A. Problem I(b). Given a matrix [email protected]?R^n^x^p and symmetric matrices [email protected]?R^p^x^p, A"[email protected]?R^r^x^r, find an nxn symmetric matrix A such [email protected]?X^[email protected]?=min,s.t. A([1,r])=A"0. Problem II. Given an nxn symmetric matrix [email protected]?, find [email protected][email protected]?S"E such [email protected][email protected][email protected][email protected][email protected]?S"[email protected][email protected][email protected]?, where S"E 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.