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Ⅰ．Matrices,Arrays and Determinants 1

1．Linear algebra 1

2．Representation by ordered sets 1

3．Total matric algebra 2

4．Diagonal and scalar matrices 5

5．Transpose．Symmetric and skew matrices 5

6．Determinants 6

7．Properties of determinants 8

8．Rank and nullity 10

9．Identities among minors 12

10．Reducibility 14

11．Arrays and determinants of higher dimension 15

12．Matrices in non-commutative systems 16

Ⅱ．The characteristic equation 17

13．The minimum equation 17

14．The characteristic equation 17

15．Determination of the minimum equation 20

16．Characteristic roots 22

17．Conjugate sets 24

18．Limits for the characteristic roots 25

19．Characteristic roots of unitary matrices 28

Ⅲ．Associated Integral Matrices 29

20．Matrices with elements in a principal ideal ring 29

21．Construction of unimodular matrices 31

22．Associated matrices 31

23．Greatest common divisors 35

24．Linear form moduls 37

25．Ideals 38

Ⅳ．Equivalence 40

26．Equivalent matrices 40

27．Invariant factors and elementary divisors 43

28．Factorization of a matrix 44

29．Polynomial domains 45

30．Equivalent pairs of matrices 48

31．Automorphic transformations 50

Ⅴ．Congruence 51

32．Matrices with elements in a principal ideal ring 51

33．Matrices with rational integral elements 54

34．Matrices with elements in a field 56

35．Matrices in an algebraically closed field 60

36．Hermitian matrices 62

37．Automorphs 65

Ⅵ．Similarity 68

38．Similar matrices 68

39．Matrices with elements in a field 69

40．WEYR＇S characteristic 73

41．Unitary and orthogonal equivalence 75

42．The structure of unitary and orthogonal matrices 78

Ⅶ．Composition of matrices 81

43．Direct sum and direct product 81

44．Product-matrices and power-matrices 85

45．Adjugates 86

Ⅷ．Matric equations 89

46．The general linear equation 89

47．Scalar equations 94

48．The unilateral equation 95

Ⅸ．Functions of Matrices 97

49．Power series in matrices 97

50．Functions of matrices 99

51．Matrices whose elements are functions of complex variables 101

52．Derivatives and integrals of matrices 102

Ⅹ．Matrices of infinite order 104

53．Infinite determinants 104

54．Infinite matrices 106

55．A matric algebra of infinite order 106

56．Bounded matrices 108

57．Matrices with a non-denumerable number of rows and colums 110