The Theory of Matrices, 1933 Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge Series, Vol. 5

I. 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.