Integral Matrices (eBook)

Front Cover 1
Integral Matrices 4
Copyright Page 5
Contents 6
Preface 14
Acknowledgments 18
Chapter I. Background Material on Rings 20
1. Principal Ideal Rings 20
2. Units 21
3. Divisibility 21
4. Congruence and Norms 22
5. The Ascending Chain Condition 23
6. Unique Factorization 24
7. Euclidean Rings 26
8. The Chinese Remainder Theorem 27
Exercises and Problems 28
Chapter II. Eqoivalence 30
1. Definition of Matrix Ring Sn, over S 30
2. Units of Sn 30
3. Definition of Equivalence 31
4. Elementary Row Operations 31
5. Completion to a Unimodular Matrix 32
6. The Hermite Normal Form 34
7. Divisor Sums 38
8. The Hermite Normal Form Class Number 38
9. An Application of the Hermite Normal Form 40
10. Left Equivalence over a Euclidean Ring 42
11. Generators of Rn' when R is Euclidean 42
12. Two-sided Equivalence 44
13. Determinantal Divisors 44
14. Multiples 44
15. The Smith Normal Form 45
16. Invariant Factors 47
17. Elementary Divisors 48
18. Divisibility Properties of the Invariant Factors 50
19. The Multiplicativity of the Smith Normal Form 52
20. The Smith Normal Form Class Number 53
21. Applications of the Smith Normal Form 54
Exercises and Problems 58
Chapter III. Similarity 60
1. Definition of Similarity 60
2. A General Result for Any Principal Ideal Ring 60
3. Similarity over a Field 62
4. Degree and Proper Degree 62
5. A Modified Euclidean Algorithm 62
6. Two Lemmas on Degree and Proper Degree 63
7. The Fundamental Theorem on Similarity over a Field 64
8. Minimal Polynomials and Nonderogatory Matrices 65
9. Companion Matrices 65
10. The Frobenius Normal Form 66
11. Jordan Matrices 67
12. The Jordan Normal Form 67
13. Criteria for Similarity to a Diagonal Matrix 68
14. Similarity over Z 68
15. Similarity to a Block Triangular Matrix 69
16. The Theorem of Latimer and MacDuffee 71
Exercises and Problems 73
Chapter IV. Congruence 75
1. Definition of Congruence 75
2. Definition of Quadratic Form 75
3. The Skew Normal Form 76
4. A General Result for Symmetric Matrices 79
5. The Connection between Symmetric Matrices and Quadratic Forms 79
6. The Set q(A) 79
7. Results on Elements Represented by Forms 80
8. Congruence over a Field 81
9. Congruence over a Field of Characteristic # 2 82
10. Congruence over an Algebraically Closed Field of Characteristic # 2 83
11. Witt’s Theorem 83
12. Two Lemmas on Finite Fields 85
13. Congruence over a Field of Finite Characteristic 85
14. Two Lemmas on Finite Fields of Characteristic 2 86
15. Congruence over Finite Fields of Characteristic 2 87
16. Ordered Fields 88
17. Sylvester’s Law of Inertia 88
18. Congruence over the Real Field 90
19. Positive Definite Matrices 90
20. Congruence over the Rational Field 90
21. Congruence over Z 91
22. The Arithmetic Minimum 91
23. The Case n = 2 91
24. The Finiteness of the Class Number for n = 2 92
25. The General Case: Hermite’s Method 92
26. The Finiteness of the Class Number for Any n 94
Exercises and Problems 96
Chapter V. Combined Similarity and Congruence 98
1. Orthogonal Equivalence 98
2. Orthogonal Equivalence over a Field of Characteristic is not equal to 2 99
3. A Lemma of Hall and Ryser 99
4. The Theorem of Hall and Ryser 100
5. The Complex Field and Unitary Equivalence 101
6. Inner Products 102
7. Length 102
8. The Gram–Schmidt Process 102
9. Normalized Vectors and Orthonormal Bases 103
10. Completion to a Unitary Matrix 104
11. Schur’s Theorem on Unitary Equivalence 105
12. Normal Matrices and Unitary Equivalence lo a Diagonal Matrix 105
13. Miscellaneous Results 106
Chapter VI. The Geometry of Numbers 107
1. Lattices 107
2. Lattice Bases 108
3. The Determinant of a Lattice 108
4. A Geometrical Consequence 108
5. Formal Properties of Lattices 109
6. Linear Transformations 109
7. Convexity and Central Symmetry 109
8. Preservation of Convexity and Central Symmetry under Linear Transformations 110
9. Volume under Linear Transformations 110
10. The Fundamental Theorem of Minkowski 110
11. Another Form of the Fundamental Theorem 112
12. Proof that an Ellipsoid Is Convex and Centrally Symmetric 112
13. Proof that the Unit Sphere Is Convex 113
14. The Volume of an Ellipsoid 113
15. An Application to the Arithmetic Minimum 114
16. Restatement of the Fundamental Theorem 114
17. Minkowski’s “Linear Forms” Theorem 115
18. Generalization to Complex Forms 116
19. An Application to Simultaneous Diophantine Approximation 116
20. Proof of the Theorem on Lattices 117
21. The Octahedron 118
22. Further Results on Linear Forms 119
23. Consequences of Interest in Algebraic Number Theory 119
24. A General Figure 120
Chapter VII. Groups of Matrices 122
1. Definition of GL(n, S), SL(n, S) 122
2. Some Properties of CL(n, S). SL(n, S) 122
3. The Centers of GL(n, S), SL(n, S) 123
4. Definition of PGL(n, S), PSL(n, S) 124
5. The Generators of SL(n, R). R a Euclidean Ring Which Is Not of Characteristic 2 124
6. The Commutator Subgroup CL’(n, R) 126
7. The Commutator Subgroup SL’(n. R) 127
8. Principal Congruence Groups 128
9. Quotient Groups and an Isomorphism 128
10. Congruence Groups 129
11. Structure theorems for Ghe(µ) 129
12. Isomorphism Theorems for G(a, ß) 131
13. Structure Theorems for G(µ) 131
14. Reduction to Prime Powers 132
15. The Groups C(p) 132
16. The Orders of GL(n, R/(µ)), SL(n, R/(µ)) 132
17. The Groups Ghe(µ) 134
18. The Congruence Groups Ghe r, r (µ) 135
19. The Order of Ghe r, s (µ)/Ghe(µ) 135
20. The Index of Ghe r, (µ) 136
21. The Structure of Ghe r, s(µ) 136
22. The Case r = s =1 138
23. Coset Representatives 138
24. An Inclusion Theorem for Ghe r, s(µ) 139
25. The Symplectic Group . = Sp(2n, R) 143
26. The Order of . when R is a Finite Field 144
27. The Center of . 144
28. The Generators of . when R is a Proper Euclidean Ring 144
29. Substitutes for the Euclidean Algorithm 145
30. A Useful Lemma for Any Principal Ideal Ring 146
31. Principal Congruence Subgroups of Ghe and an Isomorphism 149
32. Congruence Groups 149
33. Symmetric and Symplectic Matrices Modulo µ 150
34. Competing Symplectic Matrices Modulo µ to Symplectic Matrices 151
35. Various Structure Theorems 152
36. The Order of G(µ) 154
37. The Group Ghe 0(µ) 155
Exercises and Problems 156
Chapter VIII. The Classical Modular Group and Related Groups 157
1. Definition of . 157
2. Generators of . 158
3. Proof that Ghe is a Free Product 158
4. The Commutator Subgroup of . 159
5. The Kurosh Subgroup Theorem 160
6. Criterion for a Subgroup to be Free 161
7. Elements of Ghe of Finite Period 161
8. Normal Closure 161
9. The Groups Ghe n 161
10. Theorem on Normal Subgroups 162
11. The Rank of a Subgroup 163
12. Definition of Level of a Normal Subgroup 164
13. Classification by Level 164
14. Congruence Groups 165
15. Parabolic Elements 166
16. Definition of Level of Any Subgroup 166
17. Wohlfahrt’s Theorem 167
18. Consequences of Wohlfahrt’s Theorem 168
19. The Parabolic Class Number 170
20. Related Class Numbers 172
21. The Geometric Approach 173
22. Genus 175
23. Canonical Generators and Relations 176
24. Related Groups 176
25. Discrete Groups 177
26. A Class of Free, Discrete Groups 177
27. A Theorem on Free Products 181
28. A Theorem on Representations of . 181
Exercises and Problems 182
Chapter IX. Finite Matrix Groups 183
1. Reducibility and Irreducibility 183
2. Schur's Lemma 184
3. Burnside's Theorem on Irreducible Sets of Matrices 185
4. Burnside's Theorem on Groups of Finite Exponent 188
5 . Matrix Representations and Characters 189
6. Orthogonality Relationships of the First Kind 190
7. A Divisibility Theorem 191
8. Integral Matrix Groups 192
9. Lemmas on Traces 192
10. A Bound for the Order of a Finite Integral Matrix Group 194
11. Congruential Results of Minkowski 194
12. A Theorem on a Natural Homomorphism 196
13. Automorphs 197
14. The Finite Subgroups of GL(2, Z) and GL(3,Z) 198
Exercises and Problems 200
Chapter X. Circulants 201
1. The Matrix P and Its Properties 201
2. Circulants of Finite Period 202
3. The Units of R = Z(P) 203
4. The Free Rank of R 203
5. Some Consequences 205
6. Some Subgroups of R' and Their Ranks 205
7. Relations among R1, R2, R3 206
8. Definition of kn 207
9. The Group-Theoretic Meaning of k. 207
10. A Bound for kn 208
11. Proof that kn is a Power of 2 208
12. The Case n = 8 209
13. Summary of Further Results 210
14. Generalized n-Cycles 210
15. A Theorem on Congruence 212
16. Further Problems and a Theorem on Automorphs 214
17. The Automorphs of C8 215
18. A Modification of Witt's Theorem 217
19. Lower Bounds for Class Numbers 218
Exercises and Problems 218
Chapter XI. Quadratic Forms 220
1. The Sets of Matrices Pn, Pn' 220
2. An Eigenvalue Inequality 220
3. The Hermite Constant .n 221
4. An Inequality of Mordell 221
6. A Factorization of an Arbitrary Complex Matrix 224
7. Factorization of Hermitian Positive Definite Matrices 224
8. Bounds for Cofactors 226
9. Bounds for Class Numbers 227
10. Minkowski's Proof of the Finiteness of the Class Number 228
11. Some Subrings of Complex Numbers 231
12. The Norm Constant 232
13. Hermitian Positive Definite Matrices and the Generalized Hermite Constant .n(D) 233
14. Some Consequences 234
Exercises and Problems 234
References 235
Index 240