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)

Todd, one of the founders of the journal. We wish to express our appreciation

Two common properties of Z-matrices and Hermitian matrices are considered: (1) The eigenvalue interlacing property, i.e., the two smallest real eigenvalues of a matrix are interlaced by the smallest real eigenvalue of any principal submatrix of order one less. (2) The positive GLP property, i.e., if a matrix has a positive sequence of

All but one: