Solutions of the Partially Described Inverse Quadratic Eigenvalue Problem

Given k pairs of complex numbers and vectors (closed under conjugation), we consider the inverse quadratic eigenvalue problem of constructing $n\times n$ real symmetric matrices M, C, and K (with M positive definite) so that the quadratic pencil $Q(\lambda)\equiv \lambda^2M+\lambda C+K$ has the given k pairs as eigenpairs. Using various matrix decompositions, we first construct a general solution to this problem with $k\le n$. Then, with appropriate choices of degrees of freedom in the general solution, we construct several particular solutions with additional eigeninformation or special properties. Numerical results illustrating these solutions are also presented.