Integral Matrices

Name: Integral Matrices
Brand: Academic Press
Price: 54.95 EUR
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Morris Newman(Autor*in)

Unknown Author(Herausgeber*in)

Academic Press

Erschienen am 21. Juli 1972

223 Seiten

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Front Cover
Integral Matrices
Copyright Page
Contents
Preface
Acknowledgments
Chapter I. Background Material on Rings
1. Principal Ideal Rings
2. Units
3. Divisibility
4. Congruence and Norms
5. The Ascending Chain Condition
6. Unique Factorization
7. Euclidean Rings
8. The Chinese Remainder Theorem
Exercises and Problems
Chapter II. Eqoivalence
1. Definition of Matrix Ring Sn, over S
2. Units of Sn
3. Definition of Equivalence
4. Elementary Row Operations
5. Completion to a Unimodular Matrix
6. The Hermite Normal Form
7. Divisor Sums
8. The Hermite Normal Form Class Number
9. An Application of the Hermite Normal Form
10. Left Equivalence over a Euclidean Ring
11. Generators of Rn' when R is Euclidean
12. Two-sided Equivalence
13. Determinantal Divisors
14. Multiples
15. The Smith Normal Form
16. Invariant Factors
17. Elementary Divisors
18. Divisibility Properties of the Invariant Factors
19. The Multiplicativity of the Smith Normal Form
20. The Smith Normal Form Class Number
21. Applications of the Smith Normal Form
Exercises and Problems
Chapter III. Similarity
1. Definition of Similarity
2. A General Result for Any Principal Ideal Ring
3. Similarity over a Field
4. Degree and Proper Degree
5. A Modified Euclidean Algorithm
6. Two Lemmas on Degree and Proper Degree
7. The Fundamental Theorem on Similarity over a Field
8. Minimal Polynomials and Nonderogatory Matrices
9. Companion Matrices
10. The Frobenius Normal Form
11. Jordan Matrices
12. The Jordan Normal Form
13. Criteria for Similarity to a Diagonal Matrix
14. Similarity over Z
15. Similarity to a Block Triangular Matrix
16. The Theorem of Latimer and MacDuffee
Exercises and Problems
Chapter IV. Congruence
1. Definition of Congruence
2. Definition of Quadratic Form
3. The Skew Normal Form
4. A General Result for Symmetric Matrices
5. The Connection between Symmetric Matrices and Quadratic Forms
6. The Set q(A)
7. Results on Elements Represented by Forms
8. Congruence over a Field
9. Congruence over a Field of Characteristic # 2
10. Congruence over an Algebraically Closed Field of Characteristic # 2
11. Witt's Theorem
12. Two Lemmas on Finite Fields
13. Congruence over a Field of Finite Characteristic
14. Two Lemmas on Finite Fields of Characteristic 2
15. Congruence over Finite Fields of Characteristic 2
16. Ordered Fields
17. Sylvester's Law of Inertia
18. Congruence over the Real Field
19. Positive Definite Matrices
20. Congruence over the Rational Field
21. Congruence over Z
22. The Arithmetic Minimum
23. The Case n = 2
24. The Finiteness of the Class Number for n = 2
25. The General Case: Hermite's Method
26. The Finiteness of the Class Number for Any n
Exercises and Problems
Chapter V. Combined Similarity and Congruence
1. Orthogonal Equivalence
2. Orthogonal Equivalence over a Field of Characteristic is not equal to 2
3. A Lemma of Hall and Ryser
4. The Theorem of Hall and Ryser
5. The Complex Field and Unitary Equivalence
6. Inner Products
7. Length
8. The Gram-Schmidt Process
9. Normalized Vectors and Orthonormal Bases
10. Completion to a Unitary Matrix
11. Schur's Theorem on Unitary Equivalence
12. Normal Matrices and Unitary Equivalence lo a Diagonal Matrix
13. Miscellaneous Results
Chapter VI. The Geometry of Numbers
1. Lattices
2. Lattice Bases
3. The Determinant of a Lattice
4. A Geometrical Consequence
5. Formal Properties of Lattices
6. Linear Transformations
7. Convexity and Central Symmetry
8. Preservation of Convexity and Central Symmetry under Linear Transformations
9. Volume under Linear Transformations
10. The Fundamental Theorem of Minkowski
11. Another Form of the Fundamental Theorem
12. Proof that an Ellipsoid Is Convex and Centrally Symmetric
13. Proof that the Unit Sphere Is Convex
14. The Volume of an Ellipsoid
15. An Application to the Arithmetic Minimum
16. Restatement of the Fundamental Theorem
17. Minkowski's "Linear Forms" Theorem
18. Generalization to Complex Forms
19. An Application to Simultaneous Diophantine Approximation
20. Proof of the Theorem on Lattices
21. The Octahedron
22. Further Results on Linear Forms
23. Consequences of Interest in Algebraic Number Theory
24. A General Figure
Chapter VII. Groups of Matrices
1. Definition of GL(n, S), SL(n, S)
2. Some Properties of CL(n, S). SL(n, S)
3. The Centers of GL(n, S), SL(n, S)
4. Definition of PGL(n, S), PSL(n, S)
5. The Generators of SL(n, R). R a Euclidean Ring Which Is Not of Characteristic 2
6. The Commutator Subgroup CL'(n, R)
7. The Commutator Subgroup SL'(n. R)
8. Principal Congruence Groups
9. Quotient Groups and an Isomorphism
10. Congruence Groups
11. Structure theorems for Ghe(µ)
12. Isomorphism Theorems for G(a, ß)
13. Structure Theorems for G(µ)
14. Reduction to Prime Powers
15. The Groups C(p)
16. The Orders of GL(n, R/(µ)), SL(n, R/(µ))
17. The Groups Ghe(µ)
18. The Congruence Groups Ghe r, r (µ)
19. The Order of Ghe r, s (µ)/Ghe(µ)
20. The Index of Ghe r, (µ)
21. The Structure of Ghe r, s(µ)
22. The Case r = s =1
23. Coset Representatives
24. An Inclusion Theorem for Ghe r, s(µ)
25. The Symplectic Group ? = Sp(2n, R)
26. The Order of ? when R is a Finite Field
27. The Center of ?
28. The Generators of ? when R is a Proper Euclidean Ring
29. Substitutes for the Euclidean Algorithm
30. A Useful Lemma for Any Principal Ideal Ring
31. Principal Congruence Subgroups of Ghe and an Isomorphism
32. Congruence Groups
33. Symmetric and Symplectic Matrices Modulo µ
34. Competing Symplectic Matrices Modulo µ to Symplectic Matrices
35. Various Structure Theorems
36. The Order of G(µ)
37. The Group Ghe 0(µ)
Exercises and Problems
Chapter VIII. The Classical Modular Group and Related Groups
1. Definition of ?
2. Generators of ?
3. Proof that Ghe is a Free Product
4. The Commutator Subgroup of ?
5. The Kurosh Subgroup Theorem
6. Criterion for a Subgroup to be Free
7. Elements of Ghe of Finite Period
8. Normal Closure
9. The Groups Ghe n
10. Theorem on Normal Subgroups
11. The Rank of a Subgroup
12. Definition of Level of a Normal Subgroup
13. Classification by Level
14. Congruence Groups
15. Parabolic Elements
16. Definition of Level of Any Subgroup
17. Wohlfahrt's Theorem
18. Consequences of Wohlfahrt's Theorem
19. The Parabolic Class Number
20. Related Class Numbers
21. The Geometric Approach
22. Genus
23. Canonical Generators and Relations
24. Related Groups
25. Discrete Groups
26. A Class of Free, Discrete Groups
27. A Theorem on Free Products
28. A Theorem on Representations of ?
Exercises and Problems
Chapter IX. Finite Matrix Groups
1. Reducibility and Irreducibility
2. Schur's Lemma
3. Burnside's Theorem on Irreducible Sets of Matrices
4. Burnside's Theorem on Groups of Finite Exponent
5 . Matrix Representations and Characters
6. Orthogonality Relationships of the First Kind
7. A Divisibility Theorem
8. Integral Matrix Groups
9. Lemmas on Traces
10. A Bound for the Order of a Finite Integral Matrix Group
11. Congruential Results of Minkowski
12. A Theorem on a Natural Homomorphism
13. Automorphs
14. The Finite Subgroups of GL(2, Z) and GL(3,Z)
Exercises and Problems
Chapter X. Circulants
1. The Matrix P and Its Properties
2. Circulants of Finite Period
3. The Units of R = Z(P)
4. The Free Rank of R
5. Some Consequences
6. Some Subgroups of R' and Their Ranks
7. Relations among R1, R2, R3
8. Definition of kn
9. The Group-Theoretic Meaning of k.
10. A Bound for kn
11. Proof that kn is a Power of 2
12. The Case n = 8
13. Summary of Further Results
14. Generalized n-Cycles
15. A Theorem on Congruence
16. Further Problems and a Theorem on Automorphs
17. The Automorphs of C8
18. A Modification of Witt's Theorem
19. Lower Bounds for Class Numbers
Exercises and Problems
Chapter XI. Quadratic Forms
1. The Sets of Matrices Pn, Pn'
2. An Eigenvalue Inequality
3. The Hermite Constant ?n
4. An Inequality of Mordell
6. A Factorization of an Arbitrary Complex Matrix
7. Factorization of Hermitian Positive Definite Matrices
8. Bounds for Cofactors
9. Bounds for Class Numbers
10. Minkowski's Proof of the Finiteness of the Class Number
11. Some Subrings of Complex Numbers
12. The Norm Constant
13. Hermitian Positive Definite Matrices and the Generalized Hermite Constant ?n(D)
14. Some Consequences
Exercises and Problems
References
Index

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