The purpose of this book, which was first published in 1978, is to give a complete account of the theory of permanents, their history and applications. This volume was the first complete account of the theory of permanents, covering virtually the whole of the subject, a feature that no simple survey of the theory of matrices can even attempt. The work also contains many results stated without formal proofs. This book can be used as a textbook at the advanced undergraduate or graduate level. The only prerequisites are a standard undergraduate course in the theory of matrices and a measure of mathematical maturity.
Author(s): Henryk Minc, Marvin Marcus
Series: Encyclopedia of Mathematics and Its Applications 6
Publisher: Addison-Wesley
Year: 1978
Language: English
Pages: 224
City: Reading
Contents......Page 10
Preface......Page 18
1.1. Introduction......Page 20
1.2. The Originators: Binet and Cauchy......Page 21
1.3. The Continuators: Borchardt, Cayley, and the Master from Edinburgh-Sir Thomas Muir......Page 24
1.4. Renaissance of Permanents: Muir head's Theorem, Pólya's Problem, Schur's Inequality, and van der Waerden's Conjecture......Page 27
1.5. The New Era: Marvin Marcus and Company......Page 31
Problems......Page 33
2.1. Elementary Properties......Page 34
2.2. The Permanent Function as an Inner Product......Page 38
Problems......Page 45
3.1. Incidence Matrices......Page 48
3.2. Theorems of Frobenius and König......Page 50
3.3. Structure of Square (0,1)-Matrices......Page 53
3 4. (0,1)-Circulants......Page 63
Problems......Page 67
4.1. Marshall Hall's Theorem......Page 70
4.2. (0,1)-Matrices......Page 72
4.3. Fully Indecomposable (0,1)-Matrices......Page 76
4.4. Nonnegative Matrices......Page 81
4.5. Positive Semi-definite Hermitian Matrices......Page 85
Problems......Page 90
5.1. The Marcus-Newman Theory......Page 92
5.2. Properties of Minimizing Matrices......Page 100
5.3. Some Partial Results. Friedland's Theorem......Page 105
5.4. A Conjecture of Marcus and Mine......Page 110
5.5. Lower Bounds for the Permanents of Doubly Stochastic Matrices......Page 114
Problems......Page 119
6.1. From Muir to Jurkat and Ryser......Page 122
6.2. (0,1)-Matrices......Page 126
6.3. Nonnegative Matrices......Page 129
6.4. Complex Matrices......Page 132
Problems......Page 135
7.1. Binet-Minc Method......Page 138
7.2. Ryser's Method......Page 141
7.3. Comparison of Evaluation Methods......Page 143
Problems......Page 145
8.1. Other Results......Page 148
8.2. Some Applications of Permanents......Page 154
8.3. Conjectures and Unsolved Problems-Vintage 1965......Page 167
8.4. Conjectures and Unsolved Problems-A Current List......Page 173
Problems......Page 178
Bibliography......Page 180
Index to Bibliography......Page 216
Index of Notation......Page 220
Index......Page 222
