UID:
almahu_9949697716602882
Format:
1 online resource (245 p.)
ISBN:
1-281-76875-8
,
9786611768751
,
0-08-087358-8
Series Statement:
Pure and applied mathematics; a series of monographs and textbooks ; v. 45
Content:
Integral matrices
Note:
Description based upon print version of record.
,
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
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8. The Hermite Normal Form Class Number9. 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
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2. A General Result for Any Principal Ideal Ring3. 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
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1. Definition of Congruence2. 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
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15. Congruence over Finite Fields of Characteristic 216. 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
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3. A Lemma of Hall and Ryser
Additional Edition:
ISBN 0-12-517850-6
Language:
English
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