Page 11 - Determinants and Their Applications in Mathematical Physics

x Contents

3.2.1 Rejecter and Retainer Minors ............ 18
3.2.2 Second and Higher Cofactors ............. 19
3.2.3 The Expansion of Cofactors in Terms of Higher
Cofactors ........................ 20
3.2.4 Alien Second and Higher Cofactors; Sum
Formulas ........................ 22
3.2.5 Scaled Cofactors .................... 23
3.3 The Laplace Expansion .................... 25
3.3.1 A Grassmann Proof .................. 25
3.3.2 A Classical Proof ................... 27
3.3.3 Determinants Containing Blocks of Zero Elements . 30
3.3.4 The Laplace Sum Formula .............. 32
3.3.5 The Product of Two Determinants — 2 ...... 33
3.4 Double-Sum Relations for Scaled Cofactors ......... 34
3.5 The Adjoint Determinant ................... 36
3.5.1 Definition ........................ 36
3.5.2 The Cauchy Identity ................. 36
3.5.3 An Identity Involving a Hybrid Determinant . . . 37
3.6 The Jacobi Identity and Variants .............. 38
3.6.1 The Jacobi Identity — 1 ............... 38
3.6.2 The Jacobi Identity — 2 ............... 41
3.6.3 Variants ......................... 43
3.7 Bordered Determinants .................... 46
3.7.1 Basic Formulas; The Cauchy Expansion ...... 46
3.7.2 A Determinant with Double Borders ........ 49

4 Particular Determinants 51
4.1 Alternants ............................ 51
4.1.1 Introduction ...................... 51
4.1.2 Vandermondians .................... 52
4.1.3 Cofactors of the Vandermondian ........... 54
4.1.4 A Hybrid Determinant ................ 55
4.1.5 The Cauchy Double Alternant ............ 57
4.1.6 A Determinant Related to a Vandermondian . . . 59
4.1.7 A Generalized Vandermondian ............ 60
4.1.8 Simple Vandermondian Identities .......... 60
4.1.9 Further Vandermondian Identities .......... 63
4.2 Symmetric Determinants ................... 64
4.3 Skew-Symmetric Determinants ................ 65
4.3.1 Introduction ...................... 65
4.3.2 Preparatory Lemmas ................. 69
4.3.3 Pfaffians ........................ 73
4.4 Circulants ............................ 79
4.4.1 Definition and Notation ................ 79
4.4.2 Factors ......................... 79