The Theory of Matrices

InhaltsverzeichnisI. Matrices, Arrays and Determinants.1. Linear algebra.2. Representation by ordered sets.3. Total matric algebra.4. Diagonal and scalar matrices.5. Transpose. Symmetric and skew matrices.6. Determinants.7. Properties of determinants.8. Rank and nullity.9. Identities among minors.10. Reducibility.11. Arrays and determinants of higher dimension.12. Matrices in non-commutative systems.II. The characteristic equation.13. The minimum equation.14. The characteristic equation.15. Determination of the minimum equation.16. Characteristic roots.17. Conjugate sets.18. Limits for the characteristic roots.19. Characteristic roots of unitary matrices.III. Associated Integral Matrices.20. Matrices with elements in a principal ideal ring.21. Construction of unimodular matrices.22. Associated matrices.23. Greatest common divisors.24. Linear form moduls.25. Ideals.IV. Equivalence.26. Equivalent matrices.27. Invariant factors and elementary divisors.28. Factorization of a matrix.29. Polynomial domains.30. Equivalent pairs of matrices.31. Automorphic transformations.V. Congruence.32. Matrices with elements in a principal ideal ring.33. Matrices with rational integral elements.34. Matrices with elements in a field.35. Matrices in an algebraically closed field.36. Hermitian matrices.37. Automorphs.VI. Similarity.38. Similar matrices.39. Matrices with elements in a field.40. Weyr’s characteristic.41. Unitary and orthogonal equivalence.42. The structure of unitary and orthogonal matrices.VII. Composition of matrices.43. Direct sum and direct product.44. Product-matrices and power-matrices.45. Adjugates.VIII. Matric equations.46. The general linear equation.47. Scalar equations.48. The unilateral equation.IX. Functions of Matrices.49. Power series in matrices.50. Functions of matrices.51. Matrices whose elements are functions of complex variables.52. Derivatives and integrals of matrices.X. Matrices of infinite order.53. Infinite determinants.54. Infinite matrices.55. A matric algebra of infinite order.56. Bounded matrices.57. Matrices with a non-denumerable number of rows and colums.