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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. ©2006 Society for Industrial and Applied Mathematics