Matric algebra is a mathematical abstraction underlying many seemingly diverse theories. Thus bilinear and quadratic forms, linear associative algebra (hypercomplex systems), linear homogeneous trans- formations and linear vector functions are various manifestations of matric algebra. Other branches of mathematics as number theory, differential and integral equations, continued fractions, projective geometry etc. make use of certain portions of this subject. Indeed, many of the fundamental properties of matrices were first discovered in the notation of a particular application, and not until much later re- cognized in their generality. It was not possible within the scope of this book to give a completely detailed account of matric theory, nor is it intended to make it an authoritative history of the subject. It has been the desire of the writer to point out the various directions in which the theory leads so that the reader may in a general way see its extent.
While some attempt has been made to unify certain parts of the theory, in general the material has been taken as it was found in the literature, the topics discussed in detail being those in which extensive research has taken place. For most of the important theorems a brief and elegant proof has sooner or later been found. It is hoped that most of these have been incorporated in the text, and that the reader will derive as much plea- sure from reading them as did the writer.
Reihe
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Für Beruf und Forschung
Research
Illustrationen
Maße
Höhe: 244 mm
Breite: 170 mm
Dicke: 8 mm
Gewicht
ISBN-13
978-3-642-98421-1 (9783642984211)
DOI
10.1007/978-3-642-99234-6
Schweitzer Klassifikation
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.
