Determinants and Their Applications in Mathematical Physics
86 4. Particular Determinants most general centrosymmetric determinant of order 5 is of the form A 5 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ a 1 a 2 ...
4.5 Centrosymmetric Determinants 87 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ a 1 +a 5 a 2 +a 4 2 a 3 •• b 1 +b 5 b 2 +b 4 2 b 3 •• c 1 c 2 c 3 •• b 5 ...
88 4. Particular Determinants which is centrosymmetric and can therefore be expressed as the prod- uct of two determinants of lo ...
4.5 Centrosymmetric Determinants 89 The elementt 2 n− 1 does not appear inTnbut appears in the bottom right- hand corner ofBn. H ...
90 4. Particular Determinants Whent 0 =1,tr=0,r>0,T 2 n=1,En=On,An= diag[1 0 0...0]. Hence,Pn= 1 2 ,Qn= 1, and the sign ofT 2 ...
4.6 Hessenbergians 91 Ifaij= 0 whenj−i>1, the triangular array of zero elements appears in the top right-hand corner.Hncan be ...
92 4. Particular Determinants 4.6.2 A ReciprocalPowerSeries Theorem 4.21. If ∞ ∑ r=0 (−1) r ψnt r = [ ∞ ∑ r=0 φrt r ]− 1 ,φ 0 =ψ ...
4.6 Hessenbergians 93 In the next theorem,φmandψmare functions ofx. Theorem 4.22. If φ ′ m =(m+a)Fφm− 1 ,F=F(x), then ψ ′ m =(a+ ...
94 4. Particular Determinants Also, adjusting the dummy variable inS 1 and referring to (4.6.4) with n→n−1, S 1 = n− 1 ∑ i=0 (−1 ...
4.6 Hessenbergians 95 br=ar,r> 1. Bn(0) =An, B (n) ij (0) =A (n) ij . (4.6.11) Theorem 4.23. a.B ′ n =−nBn− 1. b. n ∑ r=1 A ( ...
96 4. Particular Determinants The proof of (b) follows as a corollary since, differentiating Bn by columns, B ′ n =− n ∑ r=1 B ( ...
4.7 Wronskians 97 and where φ ′ m=(m+1)φm−^1 ,φ^0 = constant, prove that A ′ n =n(n−1)An− 1. 3.Prove that n ∏ r=1 ∣ ∣ ∣ ∣ 1 arx ...
98 4. Particular Determinants Proof. Equation (4.7.1) together with its first (n−1) derivatives form a set ofnhomogeneous equati ...
4.7 Wronskians 99 Hence, W(ty 1 ,ty 2 ,...,tyn)= ∣ ∣ (tC)(tC ′ )(tC ′′ )···(tC (n−1) ) ∣ ∣ =t n ∣ ∣CC′C′′···C(n−1) ∣ ∣. The theo ...
100 4. Particular Determinants = 0 if the parameters are not distinct W ′ ijk...r = the sum of the determinants obtained by incr ...
4.7 Wronskians 101 c. ( W (n) in ) ′ =−W (n) i,n− 1. Proof. LetZidenote then-rowed column vector in which the element in rowiis ...
102 4. Particular Determinants =− 1 W 2 n ∑ s=j− 1 ,n W (n) is ∑ r wr,s+1W (n) rj , Wn ( W (n) ij )′ −W (n) ij W ′ n=−WnW (n) i, ...
4.7 Wronskians 103 Then, ε ′ rs=εr+1,s+εr,s+1 (4.7.15) and εr 0 =δr,n− 1. (4.7.16) Differentiating (4.7.16) repeatedly and apply ...
104 4. Particular Determinants = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ff ′ f ′′ ··· f (n−1) f ′ f ′′ f ′′′ ··· ··· f ′′ f ′′′ f (4) ··· ··· ··· ··· ...
4.8 Hankelians 1 105 It follows that aji=aij, so that Hankel determinants are symmetric, but it also follows that ai+k,j−k=aij,k ...
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