Article history: Received 21 April 2008 Accepted 30 March 2009 Available online 9 May 2009 Submitted by R. Brualdi AMS classification: 15A22 37J25 47A75

We present in this paper some results for solving the Quadratic Eigenvalue Complementarity Problem (QEiCP) by using DC(Difference of Convex functions) programming approaches. Two equivalent Nonconvex Polynomial Programming (NLP) formulations of QEiCP are introduced. We focus on the construction of the DC programming formulations of the QEiCP from these NLPs. The corresponding numerical solution...

Given a graph G = (V, E), we define the transpose degree sequence d T j to be equal to the number of vertices of degree at least j. We define L G , the graph Laplacian, to be the matrix, whose rows and columns are indexed by the vertex set V , whose diagonal entry at v is the degree of v and whose value at a pair (v, w) is −1 if (v, w) ∈ E and 0 otherwise. Grone and Merris conjectured [GM] Conj...

We first give the representation of the general solution of the following inverse quadratic eigenvalue problem IQEP : given Λ diag{λ1, . . . , λp} ∈ Cp×p , X x1, . . . , xp ∈ Cn×p, and both Λ and X are closed under complex conjugation in the sense that λ2j λ2j−1 ∈ C, x2j x2j−1 ∈ C for j 1, . . . , l, and λk ∈ R, xk ∈ R for k 2l 1, . . . , p, find real-valued symmetric 2r 1 -diagonal matrices M,...

In this paper we consider the robust partial quadratic eigenvalue assignment problem for second-order control systems by state feedback. To simultaneously reduce the feedback norms and the sensitivity of the close-loop eigenvalues, we formulate the problem as an unconstraint minimization problem for a new proposed cost function. An explicit analytic expression of the gradient of the cost functi...

The Partial Quadratic Eigenvalue Assignment Problem (PQEVAP) concerns the reassignment of a small number of undesirable eigenvalues of a quadratic matrix pencil, while leaving the remaining large number of eigenvalues and the corresponding eigenvectors unchanged. The problem arises in controlling undesirable resonance in vibrating structures and in stabilizing control systems. The solution of t...

This paper concerns quadratic matrix functions of the form L(λ) = Mλ2 +Dλ+K where M,D,K are Hermitian n× n matrices with M > 0. It is shown how new systems of the same type can be generated with some eigenvalues and/or eigenvectors updated and this is accomplished without “spill-over” (i.e. other spectral data remain undisturbed). Furthermore, symmetry is preserved. The methods also apply for H...

Finite element model updating of quadratic eigenvalue problems (QEPs) is proposed by Friswell, Inman and Pilkey 1998, to incorporate the measured model data into the finite element model to produce an adjusted finite element model on the damping and stiffness that closely match the experimental modal data. In this paper, we mainly develop an efficient numerical method for the finite element mod...

Quadratic pencils arise in many areas of important applications. The underlying physical systems often impose inherent structures, which include the predetermined inner-connectivity among elements within the physical system and the mandatory nonnegativity of physical parameters, on the pencils. In the inverse problem of reconstructing a quadratic pencil from prescribed eigeninformation, respect...