Sylvan Elhay

Orthogonality and partial pole assignment for the symmetric definite quadratic pencil

Abstract The eigenvectors of a symmetric matrix can be chosen to form an orthogonal set with respect to the identity and to the matrix itself. Similarly, the eigenvectors of a symmetric definite linear pencil can be chosen to be orthogonal with respect to the pair. This paper presents the three sets of matrix weights with respect to which the eigenvectors of the symmetric definite quadratic pencil are orthogonal. One of these is used to derive an explicit solution of the partial pole assignment problem by state feedback control for a control system modeled by a system of second order differential equations. The solution may be of particular interest in the stabilization and control of flexible, large space structures where only a small part of the spectrum is to be reassigned and the rest of the spectrum is required to remain unchanged.

An inverse eigenvalue problem for the symmetric tridiagonal quadratic pencil with application to damped oscillatory systems

A method is presented which constructs an n by n tridiagonal, symmetric, quadratic pencil which has its

Updating and downdating of orthogonal polynomials with data fitting applications

2n

NONSYMMETRIC LANCZOS AND FINDING ORTHOGONAL POLYNOMIALS ASSOCIATED WITH INDEFINITE WEIGHTS

eigenvalues and the

Dealing with Zero Flows in Solving the Nonlinear Equations for Water Distribution Systems

2n - 2

Jacobian Matrix for Solving Water Distribution System Equations with the Darcy-Weisbach Head-Loss Model

of its

Calculation of the weights of interpolatory quadratures

n - 1

Reformulated Co-Tree Flows Method Competitive with the Global Gradient Algorithm for Solving Water Distribution System Equations

-dimensional leading principal subpencil prescribed. It is shown that if the given eigenvalues are distinct, there are at most

An affine inverse eigenvalue problem

2^n ( 2n - 3 )!/( n - 2 )!

On Some Eigenvector-Eigenvalue Relations

different solutions. In the degenerate case, where some of the given eigenvalues are common, there are an infinite number of solutions. Apart from finding the roots of certain polynomials, the problem is solved in a finite number of steps. Where the problem has only a finite number of solutions, they can all be found in a systematic manner. The method is demonstrated with a simple example and its use is illustrated with a practical engineering application in vibrations.