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

Abstract Let R ∈ R n × n and S ∈ R n × n be nontrivial involutions, i.e.,  R = R − 1 ≠ ± I and S = S − 1 ≠ ± I . A matrix A ∈ R n × n is called ( R , S ) -symmetric if R A S = A . This paper presents a ( 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 ) -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 ) -symmetric matrix solution to the best approximation problem. Two numerical examples are given to illustrate the effectiveness of our results.

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