The inverse eigenproblem with a submatrix constraint and the associated approximation problem for (R,S)(R,S)-symmetric matrices

Let R∈Rn×nR∈Rn×n and S∈Rn×nS∈Rn×n be nontrivial involutions, i.e.,  R=R−1≠±I and S=S−1≠±I. A matrix A∈Rn×n is called (R,S)(R,S)-symmetric if RAS=ARAS=A. This paper presents a (R,S)(R,S)-symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint. The solvability condition of the constrained inverse eigenproblem is also derived. The existence, the uniqueness and the expression of the (R,S)(R,S)-symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved, respectively. An algorithm is presented to compute the (R,S)(R,S)-symmetric matrix solution to the best approximation problem. Two numerical examples are given to illustrate the effectiveness of our results.