Exact Solution of a Linear-Quadratic Inverse Eigenvalue Problem on a Certain Hamiltonian Symmetric Matrices 1, 2,

This paper investigate the exact solution of an inverse eigenvalue problem (IEP) on a certain Hamiltonian symmetric matrices namely singular symmetric matrices of rank 1 and non-singular symmetric matrices in the neighborhood of the first type of matrices via the Newton’s iterative method

Reconstruction of Structured Quadratic Pencils from Eigenvalues on Ellipses and Parabolas

In the present paper we study the reconstruction of a structured quadratic pencil from eigenvalues distributed on ellipses or parabolas. A quadratic pencil is a square matrix polynomial QP(λ) = M λ2+Cλ +K, where M, C, and K are real square matrices. The approach developed in the paper is based on the theory of orthogonal polynomials on the real line. The results can be applied to more general distribution of eigenvalues. The problem with added single eigenvector is also briefly discussed. As an illustration of the reconstruction method, the eigenvalue problem on linearized stability of certain class of stationary exact solution of the Navier-Stokes equations describing atmospheric flows on a spherical surface is reformulated as a simple mass-spring system by means of this method

Solutions to an inverse monic quadratic eigenvalue problem

AbstractGiven n+1 pairs of complex numbers and vectors (closed under complex conjugation), the inverse quadratic eigenvalue problem is to construct real symmetric or anti-symmetric matrix C and real symmetric matrix K of size n×n so that the quadratic pencil Q(λ)=λ2In+λC+K has the given n+1 pairs as eigenpairs. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are obtained. Numerical algorithms for solving the problem are developed. Numerical examples illustrating these solutions are presented

On Inverse Quadratic Eigenvalue Problems With Partially Prescribed Eigenstructure

The inverse eigenvalue problem of constructing real and symmetric square matrices M,C and K of size n n for the quadratic pencil Q(#) = # M + #C +K so that Q(#) has a prescribed subset of eigenvalues and eigenvectors is considered. This paper consists of two parts addressing two related but di#erent 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 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(#) 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 In ). It is shown via construction that the monic quadratic pencil Q(#) = In +#C+K with n+1 arbitrarily assigned complex conjugately closed pairs of distinct eigenvalues and column eigenvectors which span the space C always exists. Su#cient conditions under which this quadratic inverse eigenvalue problem is uniquely solvable are specified