A direct and simple proof of Jacobi identities for determinants
The Jacobi identities play an important role in constructing the explicit exact solutions of a broad class of integrable systems in soliton theory. In the paper, a direct and simple proof of the Jacobi identities for determinants is presented by empl…
Authors: Kuihua Yan
A DIRECT AND SIMPLE PR OOF OF JACOBI IDENTITIES F OR DETERMINANTS KUIHUA Y AN Abstra ct. The Jacobi identiti es pla y an imp ortan t role in construct- ing the explicit exact solutio ns of a broad class of in tegrable systems in soliton theory . In the pap er, a d irect and simple proof of the Ja- cobi identities for determinan ts is presen ted by employing the Pl ¨ u ck er relations. 1. Introduction Let A = a ij n × n b e a n -order matrix. Denoted by M ij R ij ≡ ( − 1) i + j M ij the cofacto r (algebraic cofactor) of the matrix en try a ij . The cofactor (alge- braic cofactor) of the minor determinan t a ij a il a kj a kl is denoted by M i j k l R i j k l ≡ ( − 1) k + l + i + j M i j k l , then the f ollo wing Jacobi iden tities [1, 2] M ii M j j − M ij M j i = M i j i j det A, 1 ≤ i, j ≤ n, (1) are v alid. Though the Jaco bi ident ities ha v e b een pro v ed in [1], as the author in [ 2] said, lo oking at the pro of o f the general case , it is difficult to understand the Jac obi ident ities immediately and the a uthor himself ca me to un derstand the result by c h ecking the formula e using computer al gebra, lo oki ng for an alternativ e pr oof and applying it to actual problems. Here w e will p resen t a direct pr oof for the Jacobi identi ties u s in g the famous Pl ¨ u c k er relations for determinant s. 2. Pl ¨ u cker rela tions In this section, le t’s state the Pl ¨ u ck er rela tions for determinan ts. Theorem 2.1. L et M b e a n × ( n − r ) matrix and a 1 , a 2 , · · · , a 2 r 2 r n -or der c olumn ve ctors, then X σ ( − 1) k 1 + ··· + k r M a k 1 · · · a k r · M a k r +2 · · · a k 2 r = 0 , (2) wher e k 1 , k 2 , · · · , k 2 r is a p ermutation of 1 , 2 , · · · , 2 r and σ is the p ermutation with 1 ≤ k 1 < · · · < k r ≤ 2 r and k r +1 < · · · < k 2 r . 1991 Mathematics Subje ct Classific ation. Primary 15A15; Secondary 11C20, 58A17. Key wor ds and phr ases. Jacobi identit y , Pl ¨ u ck er relation, Pfaffian. 1 Pr o of. Firstly , it is o bvio us that M 0 a 1 a 2 · · · a 2 r 0 M a 1 a 2 · · · a 2 r = M − M 0 0 · · · 0 0 M a 1 a 2 · · · a 2 r = M 0 0 0 · · · 0 0 M a 1 a 2 · · · a 2 r = 0 . (3) On the other hand , by the classical Laplace expansion for determinant s, it can b e obtained that M 0 a 1 a 2 · · · a 2 r 0 M a 1 a 2 · · · a 2 r = X 1 ≤ k 1 < ···
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