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The Theory of Matrices

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概要 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
目次 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.
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登録日 2020.06.27
更新日 2021.05.09