Matrix Inverse Eigenvalue Problems

Abstract

The term matrix eigenvalue problems refers to the computation of the eigenvalues of a symmetric matrix. By contrast, the term inverse matrix eigenvalue problem refers to the construction of a symmetric matrix from its eigenvalues. While matrix eigenvalue problems are well posed, inverse matrix eigenvalue problems are ill posed: there is an infinite family of symmetric matrices with given eigenvalues. This means that either some extra constraints must be imposed on the matrix, or some extra information must be supplied. These lectures cover four main areas: i) Classical inverse problems relating to the construction of a tridiagonal matrix from its eigenvalues and the first (or last) components of its eigenvectors. ii) Application of these results to the construction of simple in-line mass-spring systems, and a discussion of extensions of these results to systems with tree structure. iii) Isospectral systems (systems that all have the same eigenvalues) studied in the context of the QR algorithm, with special attention paid to the important concept of total positivity. iv) Introduction to the concept of Toda flow, a particular isospectral flow.