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.
Volume 2: XI. Complex symmetric, skew-symmetric, 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 skew-symmetric 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 non-negative elements: 1. General properties; 2. Spectral properties of irreducible non-negative 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 non-negative 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 Lappo-Danilevskii XV. The problem of Routh-Hurwitz 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 Routh-Hurwitz; 7. Orlando's formula; 8. Singular cases in the Routh-Hurwitz 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 Routh-Hurwitz theorem; 13. Some supplements to the Routh-Hurwitz theorem. Stability criterion of Lienard 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 Routh-Hurwitz problem Bibliography Index.