Given pairs of complex numbers and vectors (closed under conjugation), we consider the inverse quadratic eigenvalue problem of constructing real matrices , , , and , where , and are symmetric, and is skew-symmetric, so that the quadratic pencil has the given pairs as eigenpairs. First, we construct a general solution to this problem with . Then, with the special properties and , we construct a particular solution. Numerical results illustrate these solutions. 1. Introduction Vibrating structures such as buildings, bridges, highways, and airplanes are distributed parameter systems [1]. Very often a distributed parameter system is first discretized to a matrix second-order using techniques of finite element or finite difference, and then an approximate solution is obtained for the discretized model. Associated with the matrix second-order model is the eigenvalue problem of the quadratic pencil, where , , , and are, respectively, mass, damping, gyroscopic and stiffness matrices. The system represented by (1) is called damped gyroscopic system. In general, the gyroscopic matrix is always skew-symmetric, the damping matrix and the stiffness matrix are symmetric, the mass matrix is symmetric positive definite, and they are all real matrices. If , the system is called damped nongyroscopic system, and if , the system is called undamped gyroscopic system. The damped gyroscopic system has been widely studied in two aspects: the quadratic eigenvalue problem (QEP) and the quadratic inverse eigenvalue problem (QIEP). The QEP involves finding scalars and nonzero vectors , called the eigenvalues and eigenvectors of the system, to satisfy the algebraic equation , when the coefficient matrices are given. Many authors have been devoted to this kind of problems and a series of good results have been made (see, e.g., [2–8]). The QIEP determines or estimates the parameters of the system from observed or expected eigeninformation of . Our main interest in this paper is the corresponding inverse problem: given partially measured information about eigenvalues and eigenvectors, we reconstruct matrices , , , and , satisfied with several conditions, so that has the given eigenpairs. The problem we considered is stated as follows. Problem 1. Given an eigeninformation pair , where with find real matrices , , , and , with being symmetric definite, and being symmetric, and being skew-symmetric, so that In [9], Gohberg et al. developed a powerful GLR theory to solve the QIEP of the undamped gyroscopic system. In [10], Chu and Xu developed an elegant procedure to obtain a real-valued
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