Natalie Baddour

By Patrick Dumond and Natalie Baddour

An inverse eigenvalue problem is one where a set or subset of (generalized) eigenvalues is specified and the matrices that generate it are sought. Many methods for solving inverse eigenvalue problems are only applicable to matrices of a specific type. In this chapter, two recently proposed methods for structured (direct) solutions of inverse eigenvalue problems are presented. The presented methods are not restricted to matrices of a specific type and are thus applicable to matrices of all types. For the first method, the Cayley–Hamilton theorem is developed for the generalized eigenvalue vibration problem. For a given (desired) frequency spectrum, many solutions are possible. Hence, a discussion of the required information and suggestions for including structural constraints are given. An algorithm for solving the inverse eigenvalue problem using the generalized Cayley–Hamilton theorem is then demonstrated. An algorithm for solving partially described systems is also specified. The Cayley–Hamilton theorem algorithm is shown to be a good tool for solving inverse generalized eigenvalue problems. Examples of application of the method are given. A second method, referred to as the inverse eigenvalue determinant method, is also introduced. This method provides another direct approach to the reconstruction of the matrices of the generalized eigenvalue problem, given knowledge of its eigenvalues and various physical parameters. As for the first method, there are no restrictions on the type of matrices allowed for the inverse problem. Examples of application of the method are also given, including application-oriented examples.

Part of the book: Applied Linear Algebra in Action

By Natalie Baddour

The Hankel transform is an integral transform and is also known as the Fourier-Bessel transform. Until recently, there was no established discrete version of the transform that observed the same sort of relationship to its continuous counterpart as the discrete Fourier transform does to the continuous Fourier transform. Previous definitions of a discrete Hankel transform (DHT) only focused on methods to approximate the integrals of the continuous Hankel integral transform. Recently published work has remedied this and this chapter presents this theory. Specifically, this chapter presents a theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The standard set of shift, modulation, multiplication, and convolution rules are shown. In addition to being a discrete transform in its own right, this DHT can approximate the continuous forward and inverse Hankel transform.

Part of the book: Fourier Transforms