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The Theory of Matrices, Volume 2

F. R. Gantmacher
American Mathematical Society/Chelsea
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  • 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-Danilevskií
  • 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 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 Routh-Hurwitz problem
  • Bibliography
  • Index