## ＜電子ブック＞The Theory of Matrices

責任表示 by C. C. Mac Duffee Mac Duffee, C. C SpringerLink (Online service) English (英語) Springer Berlin Heidelberg 1933- Berlin, Heidelberg, Germany シリーズ Ergebnisse der Mathematik und Ihrer Grenzgebiete ; 5 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.続きを見る I. Matrices, Arrays and Determinants1. Linear algebra2. Representation by ordered sets3. Total matric algebra4. Diagonal and scalar matrices5. Transpose. Symmetric and skew matrices6. Determinants7. Properties of determinants8. Rank and nullity9. Identities among minors10. Reducibility11. Arrays and determinants of higher dimension12. Matrices in non-commutative systemsII. The characteristic equation13. The minimum equation14. The characteristic equation15. Determination of the minimum equation16. Characteristic roots17. Conjugate sets18. Limits for the characteristic roots19. Characteristic roots of unitary matricesIII. Associated Integral Matrices20. Matrices with elements in a principal ideal ring21. Construction of unimodular matrices22. Associated matrices23. Greatest common divisors24. Linear form moduls25. IdealsIV. Equivalence26. Equivalent matrices27. Invariant factors and elementary divisors28. Factorization of a matrix29. Polynomial domains30. Equivalent pairs of matrices31. Automorphic transformationsV. Congruence32. Matrices with elements in a principal ideal ring33. Matrices with rational integral elements34. Matrices with elements in a field35. Matrices in an algebraically closed field36. Hermitian matrices37. AutomorphsVI. Similarity38. Similar matrices39. Matrices with elements in a field40. Weyr's characteristic41. Unitary and orthogonal equivalence42. The structure of unitary and orthogonal matricesVII. Composition of matrices43. Direct sum and direct product44. Product-matrices and power-matrices45. AdjugatesVIII. Matric equations46. The general linear equation47. Scalar equations48. The unilateral equationIX. Functions of Matrices49. Power series in matrices50. Functions of matrices51. Matrices whose elements are functions of complex variables52. Derivatives and integrals of matricesX. Matrices of infinite order53. Infinite determinants54. Infinite matrices55. A matric algebra of infinite order56. Bounded matrices57. Matrices with a non-denumerable number of rows and colums.続きを見る Full text available from SpringerLink ebooks - Mathematics and Statistics (Archive)

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レコードID 3671195 QA1-939 510 Mathematics. Mathematics. Mathematics, general. ssj0001298744 9783642984211(print) 9783642992346[364299234X] 2020.06.27 2021.05.09