F. R. Gantmacher

• 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 formula8. 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