The Theory of Matrices

137 Pages·1956·10.045 MB·English

by  Cyrus Colton Mac Duffee

The theory of matrices / by C.C. MacDuffee.
MacDuffee, Cyrus Colton, 1895-
New York : Chelsea , 1956.
http://hdl.handle.net/2027/mdp.49015001327999
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ERGEBNISSE DER MATHEMATIK
UND IHRER GRENZGEBIETE
HERAUSGEGEBEN VON DER SCHRIFTLEITUNG
DES
„ZENTRALBLATT FÜR MATHEMATIK"
ZWEITER BAND
THE
THEORY
OF MATRICES
By
MACDUFFEE
C. C.
Corrected Reprint of First Edition
CHELSEA PUBLISHING COMPANY
NEW YORK, NEW YORK
PRINTED IN THE UNITED STATESOF AMERICA
85258
Preface.
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.
Acknowledgment is due Dr. Laurens Earle Bush for a critical
reading of the manuscript.
Cyrus Colton MacDuffee.
Contents.
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
Contents. V
page
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