Solutions to an inverse monic quadratic eigenvalue problem

Abstract

Given n + 1 pairs of complex numbers and vectors (closed under complex conjugation), the inverse quadratic eigenvalue problem is to construct real symmetric or anti-symmetric matrix C and real symmetric matrix K of size n × n so that the quadratic pencil Q ( λ ) = λ 2 I n + λ C + K has the given n + 1 pairs as eigenpairs. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are obtained. Numerical algorithms for solving the problem are developed. Numerical examples illustrating these solutions are presented.