the determinant business v3 REVIEWED

The determinant business PhL 2.19.16 5.17.16: The issue here is how det(RIJ) mysteriously appears when you change from ΣJ to Σ'J. This whole issue is now reduced to a small theorem in Appendix A that captures the math essence of the issue. Consider the following transformation from x-space Rn to y-space Rm , y = φ(x) (1) The differential of this equation can be written as dy = dφ(x) = (Dφ(x)) dx dyi = dφi(x) = Σj=1n (Dφ(x))ijdxj . (2) The object (Dφ(x)) is a generally non-square matrix representing the "differential" of the transformation, which we can write in several ways: (Dφ(x))ij = Rij = ∂jφi(x) = = . // R = (Dφ) (3) The most compact notation is Rij which is the "R matrix" used in Chapter 2 with x' = F(x) as the transformation notation. The R matrix has m rows (first index i going with φi) and n columns (second index j going with xj). Let k be an integer such that k ≤ min(m,n), and let dyI be an ordered wedge product of k dyi objects. For example, for k = 3 and n ≥ 5 one could have dyI = dy1^dy3^dy5 , dy1^dy2^dy5 , dy1^dy2^dy2, etc. (4) By saying dyI is "ordered" we mean that i1 < i2 < i3. In general one would write dyI = dyi^ dyi^ .....^ dyi I is ordered (5) If (2) is installed k times into (5) one obtains, dyI = dyi^ dyi^ .....^ dyi = dφi^ dφi^ .....^ dφi = Σjj...j=1n RijRij....Rij dxj^ dxj^ .....^ dxj . (6) Notice two facts about this sum. First, due to the nature of the wedge product, all non-vanishing terms in the sum must have different values of the jr indices. Second, the jr indices on dxj^ dxj^ .....^ dxj are in general not ordered in the sense defined above. We can use multiindex notation to abbreviate (6) as follows, dyI = dφI = ΣJ RIJ dxJ dyI = ordered, dxJ = not ordered (7) We want now to organize the sum over J = [j1, j2, ...jk] into a sum of sums that can be illustrated by this example with k = 3 and n = 4 : J = [j1, j2, j3] = sum 1 + sum 2 + sum 3 + sum 4 = [1,2,3] + 5 permutations // sum 1 having 3! terms + [1,2,4] + 5 permutations // sum 2 having 3! terms + [1,3,4] + 5 permutations // sum 3 having 3! terms + [2,3,4] + 5 permutations // sum 4 having 3! terms (8) In this notation we are not showing the actual terms of the sum, we are just showing the values that J takes for the terms of the sum. In this example, the n! = 4! = 24 terms in the full sum on J are broken down into four subsums each having 6 terms. Notice that the leading term in each subsum is an ordered J. The number of subsums is the number of such ordered J values which we know from (7.3.6) is (n,k), in this case (4,3) = 4. If we denote the ordered sums by Ji, for the general case one would have J = [j1, j2, ....jk] = sum 1 + sum 2 + ... + sum (n,k) = J1 + (k! - 1) permutations // sum 1 having k! terms + J2 + (k! - 1) permutations // sum 2 having k! terms + ... + J(n,k) + (k! - 1) permutations // sum (n,k) having k! terms (9) Recall our notation from *** that ΣJ is a full symmetric sum over all J values, whereas Σ'J is a sum over only ordered values of J. We can then (7) as dyI = dφI = ΣJ RIJ dxJ = Σ'J [ RIJ dxJ + (k! - 1) permutations ] (10) so the sum is now decomposed into (n,k) subsums each having k! terms. We now examine one of these subsums, using the permutation notation of Section (A.1), RIJ dxJ + (k! - 1) permutations = ΣP RIP(J) dxP(J) = ΣP RijRij....Rij dxj ^ dxj ^ .... ^ dxj . (11) From (7.3.5) with er → dxr we know that dxj ^ dxj ^ .... ^ dxj = (-1)S(P) dxj^ dxj^ .....^ dxj or dxP(J) = (-1)S(P) dxJ (12) where S(P) is the number of pairwise swaps needed to bring the permuted index P(J) back to the ordered index J. Thus we have [RIJ dxJ + (k! - 1) permutations ] = ΣP RIP(J) [(-1)S(P) dxJ ] = [ΣP (-1)S(P) RIP(J) ] dxJ . (13) But from ** we know that ΣP (-1)S(P) RIP(J) is the determinant of a square kxk portion of the full mxn R matrix whose rows are denoted by set I and whose columns are denoted by set J. We write ΣP (-1)S(P) RIP(J) = det(RIJ) (14) and then [RIJ dxJ + (k! - 1) permutations ] = det(RIJ) . (15) Using this in (10) then gives our final form dyI = dφI = ΣJ RIJ dxJ = Σ'J det(RIJ) dxJ = Σ'J det[(Dφ)IJ] dxJ (16) The object det(RIJ) is a minor of the matrix R. It is the determinant of the kxk submatrix obtained by crossing out from the full R matrix all rows other than those indicated by I, and crossing out all columns other than those indicated by J. As the derivation has shown, the determinant arises when the symmetric sum ΣJ is replaced by the ordered sum Σ'J in which all the dxJ are ordered. We now return to the example described above where n=4 and k= 3 (and m ≥ 4 say). We take I = [i1, i2, i3] = [2,3,4] just to be specific. As shown above, there are four J values in the Σ'J sum. Let's take the second J value which is J = [1,2,4] shown in (8) . The associated subsum is then [ R21 R32R44 dx1^dx2^dx4 + 5 permutations ] = R21 R32R44 dx1^dx2^dx4 + R21 R34R42 dx1^dx4^dx2 + R24 R31R42 dx4^dx1^dx2 + R24 R32R41 dx4^dx2^dx1 + R22 R34R41 dx2^dx4^dx1 + R22 R31R44 dx2^dx1^dx4 . (17) Here we show in red the two indices which will be swapped to make the next term. The next step is to write each wedge product in terms of the ordered product dx1^dx2^dx4. A minus sign appears if an odd number of swaps is required. It is easy to show that forward cyclic gives a plus, but reverse cyclic gives a minus. Continuing the above = R21 R32R44 (dx1^dx2^dx4) + R21 R34R42 (-dx1^dx2^dx4) + R24 R31R42 (+dx1^dx2^dx4) + R24 R32R41 (-dx1^dx2^dx4) + R22 R34R41 (+dx1^dx2^dx4) + R22 R31R44 (-dx1^dx2^dx4) = [ R21 R32R44 - R21 R34R42 + R24 R31R42 - R24 R32R41 + R22 R34R41 - R22 R31R44](dx1^dx2^dx4) 1 2 3 4 5 6 (18) Using the formula (16) for our term with I = [2,3,4] and J = [1,2,4] Maple computes for us, (19) The matrix RIJ is the 3x3 portion of the full R matrix which has rows 2,3,4 and columns 1,2,4. Apart from a reordering of the terms, the formula (16) gives the same result as the manual calculation for this one of the four terms in the sum Σ'J det(RIJ) dxJ .