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Determinants and Their Applications in Mathematical Physics Robert Vein Paul Dale Springer [...]... The order and structure of rejecter minors depends on the value of n but the order and structure of retainer minors are independent of n provided only that n is sufficiently large For this reason, the parameter n has been omitted from N Examples Nip = aip Nij,pq = aip ajp aip Nijk,pqr = ajp akp 1 = aip , n ≥ 1, aiq , ajq n ≥ 2, aiq ajq akq air ajr , akr n ≥ 3 Both rejecter and retainer minors arise in. .. construction of the Laplace expansion of a determinant (Section 3.3) Exercise Prove that Nij,pq Nik,pq 3.2.2 Nij,pr = Nip Nijk,pqr Nik,pr Second and Higher Cofactors (n) The first cofactor Aij is defined in Chapter 1 and appears in Chapter 2 It is now required to generalize that concept 20 3 Intermediate Determinant Theory In the definition of rejecter and retainer minors, no restriction is made concerning... and Higher Minors and Cofactors 19 called a rejecter minor The numbers is and js are known respectively as row and column parameters Now, let Ni1 i2 ir ;j1 j2 jr denote the subdeterminant of order r which is obtained from An by retaining rows i1 , i2 , , ir and columns j1 , j2 , , jr and rejecting the other rows and columns Ni1 i2 ir ;j1 j2 jr may conveniently be called a retainer minor Examples... then Ap1 q1 Ap2 q2 = Ap2 q1 Ap1 q2 , (2.3.16) that is, Ap1 q1 Ap2 q1 Ap1 q2 = 0, Ap2 q2 1 ≤ p1 , p2 , q1 , q2 ≤ n It follows that Ap1 q1 Ap2 q1 Ap3 q1 Ap1 q2 Ap2 q2 Ap3 q2 Ap1 q3 Ap2 q3 = 0 Ap3 q2 since the second-order cofactors of the elements in the last (or any) row are all zero Continuing in this way, Ap1 q1 Ap2 q1 ··· Apr q1 Ap1 q2 Ap2 q2 ··· Apr q2 · · · Ap1 qr · · · Ap2 qr ··· ··· · · · Apr qr... to Row and Column Operations The row operations 3 Ri = uij Rj , j=i uii = 1, 1 ≤ i ≤ 3; uij = 0, i > j, (2.3.1) 2.3 Elementary Formulas 11 namely R1 = R1 + u12 R2 + u13 R3 R2 = R2 + u23 R3 R3 = R3 , can be expressed in the form R1 1 u12 R2 = 1 R3 R1 u13 u23 R2 1 R3 Denote the upper triangular matrix by U3 These operations, when performed in the given order on an arbitrary determinant... them, and few proofs are given If further proofs are required, they can be found in numerous undergraduate textbooks Several of the relations, including Cramer’s formula and the formula for the derivative of a determinant, are expressed in terms of column vectors, a notation which is invaluable in the description of several analytical processes 2.2 Row and Column Vectors Let row i (the ith row) and. .. correspond to the (n − 1) possible ways of expanding a subdeterminant of order (n − 1) by elements from one row and their cofactors Omitting the parameter n and referring to (2.3.10 ), it follows that if i < j and p < q, then Aij,pq = = ∂Aip ∂ajq ∂2A ∂aip ∂ajq (3.2.4) which can be regarded as an alternative definition of the second cofactor Aij,pq Similarly, n (n) Aij,pq (n) = 1 ≤ k ≤ n, akr Aijk,pqr ,. .. identity, and in Section 5.4.1 on the Matsuno determinant The expansion of an rth cofactor, a subdeterminant of order (n − r) , can be expressed in the form n (n) Ai1 i2 ir ;j1 j2 jr = (n) q=1 apq Ai1 i2 ir p; j1 j2 jr q , 1 ≤ p ≤ n, p = is , (3.2.7) 1 ≤ s ≤ r The r terms in which q = js , 1 ≤ s ≤ r, are zero by the first convention for cofactors Hence, the sum contains (n − r) nonzero terms, as expected... generalize the concept of first minors as defined in Chapter 1 Let An = |aij |n , and let {is } and {js }, 1 ≤ s ≤ r ≤ n, denote two independent sets of r distinct numbers, 1 ≤ is and js ≤ n Now let (n) Mi1 i2 ir ;j1 j2 jr denote the subdeterminant of order (n − r) which is obtained from An by rejecting rows i1 , i2 , , ir and columns j1 , j2 , , jr (n) Mi1 i2 ir ;j1 j2 jr is known as an rth minor... chapters However, many properties of particular determinants can be proved by performing a sequence of row and column operations and in these applications, the symbols Ri and Cj appear with equal frequency If every element in Cj is multiplied by the scalar k, the resulting vector is denoted by kCj : kCj = ka1j ka2j ka3j · · · kanj T If k = 0, this vector is said to be zero or null and is denoted by . as row and column vectors, respectively. Determinants, their elements, their rejecter and retainer minors, their simple and scaled cofactors, their row and. programming language Formula on a Macintosh computer in camera-ready form. Birmingham, U.K. P .R. Vein P. Dale Contents Preface v 1 Determinants, First Minors,