Matrix analysis

Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.

Abstract: 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.

Abstract: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography Index.