Determinants and Their Applications in Mathematical Physics Robert Vein Paul Dale Springer [...]... 5.5.3 Further Determinantal Formulas 5.6 Distinct Matrices with Nondistinct Determinants 5.6.1 Introduction 5.6.2 Determinants with Binomial Elements 5.6.3 Determinants with Stirling Elements 5.7 The One-Variable Hirota Operator 5.7.1 Definition and Taylor Relations 5.7.2 A Determinantal Identity 5.8 Some Applications of... that an arbitrary determinant An = |aij |n can be expressed in the form 1 An = n 3.2 3.2.1 n sj Sj (Trahan) j=1 Second and Higher Minors and Cofactors Rejecter and Retainer Minors It is required to 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... regard Mij and Aij as quantities which belong to aij in order to give meaning to the phrase “an element and its cofactor.” The expansion of A by elements from row i and their cofactors is n A= aij Aij , 1 ≤ i ≤ n (2.3.8) j=1 The expansion of A by elements from column j and their cofactors is obtained by summing over i instead of j: n A= aij Aij , 1 ≤ j ≤ n (2.3.9) i=1 Since Aij belongs to but is independent... element aij in the determinant A = |aij |n , there is associated a subdeterminant of order (n − 1) which is obtained from A by deleting row i and column j This subdeterminant is known as a first minor of A and is denoted by Mij The first cofactor Aij is then defined as a signed first minor: Aij = (−1)i+j Mij (2.3.7) It is customary to omit the adjective first and to refer simply to minors and cofactors and it... rejecter and retainer minors arise in the 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... Toeplitz Elements Hessenberg Determinants with Prime Elements Bordered Yamazaki–Hori Determinants — 2 Determinantal Identities Related to Matrix Identities Applications of Determinants in Mathematical Physics 6.1 Introduction 6.2 Brief Historical Notes 6.2.1 The Dale Equation 6.2.2 The Kay–Moses Equation... downward in a cyclic manner, the last element in Cj appearing as the first element in C∗ , that is, j C∗ = anj a1j a2j · · · an−1,j j T In this particular case, Theorem 3.1 can be expressed in words as follows: Theorem 3.1a Given an arbitrary determinant An , form n other determinants by dislocating the elements in the jth column one place downward in a cyclic manner, 1 ≤ j ≤ n Then, the sum of the n determinants. .. = j =1 ki = j Referring to the definition of a determinant in (1.2.4), it is seen that Aij is the determinant obtained from |aij |n by replacing row i by the row [0 0 1 0 0], where the element 1 is in column j Aij is known as the cofactor of the element aij in An Comparing (1.3.10) and (1.3.11), Aij = (−1)i+j Mij (1.3.12) (n) (n) Minors and cofactors should be written Mij and Aij but the parameter... 3.3.3 Determinants Containing Blocks of Zero Elements 3.3.4 The Laplace Sum Formula 3.3.5 The Product of Two Determinants — 2 Double-Sum Relations for Scaled Cofactors The Adjoint Determinant 3.5.1 Definition 3.5.2 The Cauchy Identity 3.5.3 An Identity Involving a Hybrid Determinant The Jacobi Identity and Variants... replacing Cj by Cj is called a column operation and is extensively applied to transform and evaluate determinants Row and column operations are of particular importance in reducing the order of a determinant Exercise If the determinant An = |aij |n is rotated through 90◦ in the clockwise direction so that a11 is displaced to the position (1, n), a1n is displaced to the position (n, n), etc., and the . manuscript in Microsoft Word 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, and Cofactors. and column parameters, respec- tively, in passages where a large number of such parameters are required. Matrices are seldom required, but where they are indispensable, they ap- pear in boldface. elements, their rejecter and retainer minors, their simple and scaled cofactors, their row and column vectors, and their derivatives have all been expressed in a notation which we believe is simple and