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The inverse eigenvalue problem of constructing real and symmetric square matrices $M$, $C$, and $K$ of size $n \times n$ for the quadratic pencil $Q(\lambda) = \lambda^2 M + \lambda C + K$ so that $Q(\lambda)$ has a prescribed subset of eigenvalues and eigenvectors is considered. This paper consists of two parts addressing two related but different problems.

The first part deals with the inverse problem where $M$ and $K$ are required to be positive definite and semidefinite, respectively. It is shown via construction that the inverse problem is solvable for any $k$, given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided $k \leq n$. The construction also allows additional optimization conditions to be built into the solution so as to better refine the approximate pencil. The eigenstructure of the resulting $Q(\lambda)$ is completely analyzed.

The second part deals with the inverse problem where $M$ is a fixed positive definite matrix (and hence may be assumed to be the identity matrix $I_n$). It is shown via construction that the monic quadratic pencil $Q(\lambda)=\lambda^2 I_n + \lambda C + K$, with $n + 1$ arbitrarily assigned complex conjugately closed pairs of distinct eigenvalues and column eigenvectors which span the space $\mathbb{C}^n$, always exists. Sufficient conditions under which this quadratic inverse eigenvalue problem is uniquely solvable are specified.