The inverse eigenvalue problem of constructing real and symmetric square matrices M , C, and K of size n × n for the quadratic pencil Q(λ) = λ2M + λC + K so that Q(λ) 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 ...

We present a Krylov subspace–type projection method for a quadratic matrix polynomial λ2I − λA − B that works directly with A and B without going through any linearization. We discuss a special case when one matrix is a low rank perturbation of the other matrix. We also apply the method to solve quadratically constrained linear least squares problem through a reformulation of Gander, Golub, and...

Motivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field ~ H. We prove the following dichotomy: the number of conjugate times is identically zero or grows to infinity. The latter case occurs if and only if ~...

A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius prope...

We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and parameterized nonlinear wave equations. The aim of the paper is to present and analyze a continuation method for invariant subspaces that belong to a group of eigenvalues the number of which is much sm...

We consider numerical methods for the computation of the eigenvalues of the tridiagonal hyperbolic quadratic eigenvalue problem. The eigenvalues are computed as zeros of the characteristic polynomial using the bisection, Laguerre’s method, the Ehrlich–Aberth method, and the Durand–Kerner method. Initial approximations are provided by a divide-and-conquer approach using rank two modifications. T...

In this paper, we continue our paper [7] to develop an efficient numerical algorithm for the finite element model updating of quadratic eigenvalue problems (QEPs). This model updating of QEPs is proposed to incorporate the measured model data into the finite element model to produce an adjusted finite element model on the mass, damping and stiffness that closely match the experimental modal dat...

In the vibration analysis of high speed trains arises such a palindromic quadratic eigenvalue problem (PQEP) (λ2AT + λQ + A)z = 0, where A, Q ∈ Cn×n have special structures: both Q and A are m ×m block matrices with each block being k × k (thus n = m × k), and Q is complex symmetric and tridiagonal block-Toeplitz, and A has only one nonzero block in the (1,m)th block position which is the same ...

The inverse eigenvalue problem of constructing square matrices M,C and K of size n for the quadratic pencil Q(λ) ≡ λM + λC +K so that Q(λ) has a prescribed subset of eigenvalues and eigenvectors is considered. This paper offers a constructive proof showing that, given any k ≤ n distinct eigenvalues and linearly independent eigenvectors, the problem is solvable even under the restriction that M,...