Determinants and Their Applications in Mathematical Physics
6 1. Determinants, First Minors, and Cofactors = n ∑ j=1 cijej, where cij= n ∑ k=1 aikbkj. (1.4.2) Hence, x 1 x 2 ···xn=|cij|ne ...
2 A Summary of Basic Determinant Theory 2.1 Introduction This chapter consists entirely of a summary of basic determinant theory ...
8 2. A Summary of Basic Determinant Theory An= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ R 1 R 2 R 3 Rn ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣C 1 C 2 C 3 ···Cn ∣ ∣ ...
2.3 Elementary Formulas 9 Example. ∣ ∣ C 1 C 3 C 4 C 2 ∣ ∣ =− ∣ ∣ C 1 C 2 C 4 C 3 ∣ ∣ = ∣ ∣ C 1 C 2 C 3 C 4 ∣ ∣ . Applying this ...
10 2. A Summary of Basic Determinant Theory = m ∑ k 1 =1 m ∑ k 2 =1 ··· m ∑ kn=1 ∣ ∣ C 1 k 1 ···Cjkj···Cnkn ∣ ∣ n . The function ...
2.3 Elementary Formulas 11 namely R ′ 1 =R^1 +u^12 R^2 +u^13 R^3 R ′ 2 = R^2 +u^23 R^3 R ′ 3 = R^3 , can be expressed in the for ...
12 2. A Summary of Basic Determinant Theory Similarly, the column operations C ′ i= i ∑ j=1 vijCj,vii=1, 1 ≤i≤ 3 ,vij=0, i>j, ...
2.3 Elementary Formulas 13 The elements come from rowiofA, but the cofactors belong to the elements in rowkand are said to be al ...
14 2. A Summary of Basic Determinant Theory = 1 A n ∑ i=1 biAij. (2.3.14) The solution of the triangular set of equations i ∑ j= ...
2.3 Elementary Formulas 15 2.3.6 The Cofactors of a Zero Determinant........ IfA= 0, then Ap 1 q 1 Ap 2 q 2 =Ap 2 q 1 Ap 1 q 2 , ...
3 Intermediate Determinant Theory 3.1 Cyclic Dislocations and Generalizations........... Define column vectorsCjandC ∗ j as foll ...
3.1 Cyclic Dislocations and Generalizations 17 Hence n ∑ j=1 ∣ ∣C 1 C 2 ···C ∗ j ···Cn ∣ ∣= n ∑ i=1 n ∑ r=1 (1−δir)λir n ∑ j=1 a ...
18 3. Intermediate Determinant Theory Exercises 1.Letδ r denote an operator which, when applied toCj, has the effect of dislocat ...
3.2 Second and Higher Minors and Cofactors 19 called arejecterminor. The numbersisandjsare known respectively as row and column ...
20 3. Intermediate Determinant Theory In the definition of rejecter and retainer minors, no restriction is made concerning the r ...
3.2 Second and Higher Minors and Cofactors 21 is possible to expandA (n) ip by elements from any row or column and second cofact ...
22 3. Intermediate Determinant Theory The (n−r) values ofpfor which the expansion is valid correspond to the (n−r) possible ways ...
3.2 Second and Higher Minors and Cofactors 23 which can be abbreviated with the aid of the Kronecker delta function [Appendix A] ...
24 3. Intermediate Determinant Theory etc. In simple algebraic relations such as Cramer’s formula, the advantage of using scaled ...
3.3 The Laplace Expansion 25 is applied in Section 3.6.2 on the Jacobi identity. Formulas (3.2.16) and (3.2.17) are applied in S ...
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