Fifty years after the original Russian Edition, this classic work is finally available in English for the general mathematical audience. This book lays the foundation of what later became "Krein's Theory of String". The original ideas stemming from mechanical considerations are developed with exceptional clarity. A unique feature is that it can be read profitably by both research mathematicians and engineers. The authors study in depth small oscillations of onedimensional continua with a finite or infinite number of degrees of freedom. They single out an algebraic property responsible for the qualitative behavior of eigenvalues and eigenfunctions of onedimensional continua and introduce a subclass of totally positive matrices, which they call oscillatory matrices, as well as their infinitedimensional generalization and oscillatory kernels. Totally positive matrices play an important role in several areas of modern mathematics, but this book is the only source that explains their simple and intuitively appealing relation to mechanics. There are two supplements contained in the book, "A Method of Approximate Calculation of Eigenvalues and Eigenvectors of an Oscillatory Matrix", and Krein's famous paper which laid the groundwork for the broad research area of the inverse spectral problem: "On a Remarkable Problem for a String with Beads and Continued Fractions of Stieltjes". The exposition is selfcontained. The first chapter presents all necessary results (with proofs) on the theory of matrices which are not included in a standard linear algebra course. The only prerequisite in addition to standard linear algebra is the theory of linear integral equations used in Chapter 5. The book is suitable for graduate students, research mathematicians and engineers interested in ordinary differential equations, integral equations, and their applications. Readership Graduate students, research mathematicians, and engineers interested in ordinary differential equations, integral equations, and their applications. Reviews From a review of the Russian edition: "The authors develop in this significant book an extensive theory relating largely to sets of characteristic functions ... The book is characterized throughout by a clear style, by a wealth of results, and by a close union between the mathematical and the dynamical aspects of the investigation."  Mathematical Reviews Table of Contents  Introduction
 Review of matrices and quadratic forms
 Oscillatory matrices
 Small oscillations of mechanical systems with \(n\) degrees of freedom
 Small oscillations of mechanical systems with an infinite number of degrees of freedom
 Signdefinite matrices
 A method of approximate calculation of eigenvalues and eigenvectors of an oscillatory matrix
 On a remarkable problem for a string with beads and continued fractions of Stieltjes
 Remarks
 References
 Index
