In this paper we consider the following problem: Given two sets of distinct numbers {λ_k}_(k=1)~(2n) and {μ_k}_(k=1)~(2n-2), determine tridiagonal symmetric C and K which are such that the quadratic pencil Q(λ)=λ~2I+λC + K (1) has eigenvalues {薩k}_(k=1)~(2n) and its dimension n — 1 leading principal subpencil Q(λ) has eigenvalues {μ_k}_(k=1)~(2n-2). This problem has application in the identification and construction of the most basic mechanical design components, the mass-spring-damper system. The solution to our problem is equivalent to constructing such a system with its poles and zeros prescribed. We will also mention the related problem in which C is replaced by the sum of a diagonal and a tridiagonal skew symmetric matrices. This models a certain damped system with gyroscopic forces.
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