BOOK REVIEW FOR BULLETIN OF THE AMS
Nicholas John Higham, Lloyd N. Trefethen, Mark Embree · 2006
Eigenvalues, latent roots, proper values, characteristic values—four synonyms for a set of numbers that provide much useful information about a matrix or operator. A huge amount of research has been directed at the theory of eigenvalues (localization, perturbation, canonical forms, . . . ), at applications (ubiquitous), and at numerical computation. I would like to begin with a very selective description of some historical aspects of these topics, before moving on to pseudoeigenvalues, the subject of the book under review. Back in the 1930s, Frazer, Duncan, and Collar of the Aerodynamics Department of the National Physical Laboratory (NPL), England, were developing matrix methods for analyzing flutter (unwanted vibrations) in aircraft. This was the beginning of what became known as matrix structural analysis [9], and led to the authors’ book Elementary Matrices and Some Applications to Dynamics and Differential Equations, published in 1938 [10], which was “the first to employ matrices as an engineering tool” [2]. Olga Taussky worked in Frazer’s group at NPL during the Second World War, analyzing 6× 6 quadratic eigenvalue problems (QEPs) (λ2A2 + λA1 + A0)x = 0 arising in flutter analysis of supersonic aircraft [25]. Subsequently Peter Lancaster, working at the English Electric Company in the 1950s solved QEPs of dimension 2 to 20 [12]. Taussky (at Caltech) and Lancaster (at the University of Calgary) both went on to make fundamental contributions to matrix theory, and in particular to matrix eigenvalue problems [24], [17]. So aerodynamics provided the impetus for some significant work on the theory and computation of matrix eigenvalues. In those early days efficient and reliable numerical methods for solving eigenproblems were not available. Today they are, but aerodynamics and other areas of engineering continue to provide challenges concerning eigenvalues. The trend towards extreme designs, such as high speed trains [15], micro-electromechanical (MEMS) systems, and “superjumbo” jets such as the Airbus 380, make the analysis and computation of resonant frequencies of these structures difficult [21], [26]. Extreme designs often lead to eigenproblems with poor conditioning, while the physics of the systems leads to algebraic structure