Inverse problems for nonsymmetric matrices with a submatrix constraint

In this paper the following problems are considered:Problem I(a):Given matrices X∈Rn×pX∈Rn×p with full column rank, B∈Rp×pB∈Rp×p and A0∈Rr×rA0∈Rr×r, find a matrix A∈Rn×nA∈Rn×n such thatXTAX=B,A([1,r])=A0, where A([1,r])A([1,r]) is the r×rr×r leading principal submatrix of the matrix A.Problem I(b):Given matrices X∈Rn×pX∈Rn×p, B∈Rp×pB∈Rp×p and A0∈Rr×rA0∈Rr×r, find a matrix A∈Rn×nA∈Rn×n such that‖XTAX-B‖=mins.t.A([1,r])=A0.Problem II:Given a matrix A∼∈Rn×n with A∼([1,r])=A0, find A^∈SE such that‖A∼-A^‖=infA∈SE‖A∼-A‖,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 solution of Problem I are derived. The expression of the solution of Problem II is presented. A numerical algorithm for solving the problems is provided.