AMS Chelsea Publishing 1959; 276 pp; hardcover Volume: 133 Reprint/Revision History: first AMS printing 2000 ISBN10: 0821826646 ISBN13: 9780821826645 List Price: US$43 Member Price: US$38.70 Order Code: CHEL/133.H
This item is also sold as part of the following set: CHELGANTSET  This treatise, by one of Russia's leading mathematicians, gives in easily accessible form a coherent account of matrix theory with a view to applications in mathematics, theoretical physics, statistics, electrical engineering, etc. The individual chapters have been kept as far as possible independent of each other, so that the reader acquainted with the contents of Chapter 1 can proceed immediately to the chapters of special interest. Much of the material has been available until now only in the periodical literature. Reviews "This is an excellent and unusual textbook on the application of the theory of matrices. In spite of intensive developments in the theory of matrices and appearance of other significant books, both general and specialized, in the last four decades, this monograph has retained its leading role. This textual matter includes many chapters of interest to applied mathematicians."  Zentralblatt MATH From a review of the original Russian edition ... "The first part (10 chapters; "General theory") gives in satisfactory detail, and with more than customary completeness, the topics which belong to the main body of the ... subjects ... The point of view is broad and includes much abstract treatment ... "The number of subjects which the book treats well is great ... would appeal to a wide audience."  Mathematical Reviews From a review of the English translation ... "The work is an outstanding contribution to matrix theory and contains much material not to be found in any other text."  Mathematical Reviews Table of Contents Volume 2  XI. Complex symmetric, skewsymmetric, and orthogonal matrices: 1. Some formulas for complex orthogonal and unitary matrices; 2. Polar decomposition of a complex matrix; 3. The normal form of a complex symmetric matrix; 4. The normal form of a complex skewsymmetric matrix; 5. The normal form of a complex orthogonal matrix
 XII. Singular pencils of matrices: 1. Introduction; 2. Regular pencils of matrices; 3. Singular pencils. The reduction theorem; 4. The canonical form of a singular pencil of matrices; 5. The minimal indices of a pencil. Criterion for strong equivalence of pencils; 6. Singular pencils of quadratic forms; 7. Application to differential equations
 XIII. Matrices with nonnegative elements: 1. General properties; 2. Spectral properties of irreducible nonnegative matrices; 3. Reducible matrices; 4. The normal form of a reducible matrix; 5. Primitive and imprimitive matrices; 6. Stochastic matrices; 7. Limiting probabilities for a homogeneous Markov chain with a finite number of states; 8. Totally nonnegative matrices; 9. Oscillatory matrices
 XIV. Applications of the theory of matrices to the investigation of systems of linear differential equations: 1. Systems of linear differential equations with variable coefficients. General concepts; 2. Lyapunov transformations; 3. Reducible systems; 4. The canonical form of a reducible system. Erugin's theorem; 5. The matricant; 6. The multiplicative integral. The infinitesimal calculus of Volterra; 7. Differential systems in a complex domain. General properties; 8. The multiplicative integral in a complex domain; 9. Isolated singular points; 10. Regular singularities; 11. Reducible analytic systems; 12. Analytic functions of several matrices and their application to the investigation of differential systems. The papers of LappoDanilevskií
 XV. The problem of RouthHurwitz and related questions: 1. Introduction; 2. Cauchy indices; 3. Routh's algorithm; 4. The singular case. Examples; 5. Lyapunov's theorem; 6. The theorem of RouthHurwitz; 7. Orlando's formula; 8. Singular cases in the RouthHurwitz theorem; 9. The method of quadratic forms. Determination of the number of distinct real roots of a polynomial; 10. Infinite Hankel matrices of finite rank; 11. Determination of the index of an arbitrary rational fraction by the coefficients of numerator and denominator; 12. Another proof of the RouthHurwitz theorem; 13. Some supplements to the RouthHurwitz theorem. Stability criterion of Liénard and Chipart; 14. Some properties of Hurwitz polynomials. Stieltjes' theorem. Representation of Hurwitz polynomials by continued fractions; 15. Domain of stability. Markov parameters; 16. Connection with the problem of moments; 17. Theorems of Markov and Chebyshev; 18. The generalized RouthHurwitz problem
 Bibliography
 Index
