Determinants and Their Applications in Mathematical Physics
26 3. Intermediate Determinant Theory Recalling the definitions of rejecter minorsM, retainer minorsN, and cofactorsA, each with ...
3.3 The Laplace Expansion 27 Whenr=2, An= ∑ Nir,jsAir,js, summed overi,rorj, s, = ∑ ∣ ∣ ∣ ∣ aij ais arj ars ∣ ∣ ∣ ∣ Air,js. 3.3. ...
28 3. Intermediate Determinant Theory Substituting the first line of (3.3.9 and the second line of (3.3.8), A= n ∑ i 1 =1 n ∑ i ...
3.3 The Laplace Expansion 29 The expansion formula (3.3.4) follows. Illustrations 1.Whenr= 2, the Laplace expansion formula can ...
30 3. Intermediate Determinant Theory Now, interchange the dummies wherever necessary in order thatp< q<rin all sums. The ...
3.3 The Laplace Expansion 31 The retainer minor is signless and equal toR. The sign of the cofactor is (−1) k , where k= n ∑ r=1 ...
32 3. Intermediate Determinant Theory Example 3.3. Let V 2 n= ∣ ∣ ∣ ∣ ∣ ∣ Eip Fiq Giq Eip Giq Fiq Ojp Hjq Kjq ∣ ∣ ∣ ∣ ∣ ∣ 2 n , ...
3.3 The Laplace Expansion 33 Exercises 1.Ifn= 4, prove that ∑ p<q N 23 ,pqA 24 ,pq= ∣ ∣ ∣ ∣ ∣ ∣ ∣ a 11 a 12 a 13 a 14 a 21 a ...
34 3. Intermediate Determinant Theory The result is: AnBn= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ c 11 c 12 ... c 1 n c 21 c 22 ... c 2 n ...
3.4 Double-Sum Relations for Scaled Cofactors 35 A ′ and (A ij ) ′ and the other two are identities: A ′ A = (logA) ′ = n ∑ r=1 ...
36 3. Intermediate Determinant Theory = ∑ r frA rj δri+ ∑ s gsA is δsj =fiA ij +gjA ij which proves (D). The proof of (C) is sim ...
3.5 The Adjoint Determinant 37 Proof. AadjA=|aij|n|Aji|n =|bij|n, where, referring to Section 3.3.5 on the product of two determ ...
38 3. Intermediate Determinant Theory Hence, removing the factorAnfrom each row, |cij|n=A n n ∣ ∣ ∣ ∣ δijxi+ Hij An ∣ ∣ ∣ ∣ n wh ...
3.6 The Jacobi Identity and Variants 39 It is required to prove that J 12 ...r;12...r=A r− 1 M 12 ...r;12...r =A r− 1 A 12 ...r; ...
40 3. Intermediate Determinant Theory as a block in the top left-hand corner. Denote the result by (adjA) ∗ . Then, (adjA) ∗ =σa ...
3.6 The Jacobi Identity and Variants 41 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ Aa 21 a 23 Aa 31 a 33 a 11 a 13 a 41 a 43 ∣ ∣ ∣ ∣ ∣ ∣ ∣ =A 2 ∣ ∣ ∣ ∣ a 1 ...
42 3. Intermediate Determinant Theory =− ∑ r ∑ s δrpδsqA is A rj =−A iq A pj =− 1 A 2 [AiqApj]. (3.6.7) Hence, ∣ ∣ ∣ ∣ Aij Aiq A ...
3.6 The Jacobi Identity and Variants 43 But also, ∂ 3 A ∂aij∂apq∂auv =Aipu,jqv. (3.6.11) Hence, ∣ ∣ ∣ ∣ ∣ ∣ Aij Aiq Aiv Apj Apq ...
44 3. Intermediate Determinant Theory Proof. Denote the left side of variant (A) byE. Then, applying the Jacobi identity, An+1E= ...
3.6 The Jacobi Identity and Variants 45 ButA (n+1) i,n+1;j,n+1=A (n) ij. Hence,AnP= 0. The result follows. Three particular ca ...
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