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)

Abstract. We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficients αn = �L ℓ=1 Cℓbn ℓ + O((b∆) n) where ∆ < 1 and |bℓ | = b < 1. 1.

Abstract. For the standard symplectic forms on Jacobi and CMV matrices, we compute Poisson brackets of OPRL and OPUC, and relate these to other basic Poisson brackets and to Jacobians of basic changes of variable. 1.

Abstract. We give a short review of the spectral instability of non-normal semiclassical differential operators, both for scalar operators and systems. 1.