18.325: Finite Random Matrix TheoryJacobians of Matrix Transforms (with wedge products)Professor Alan EdelmanHandout #3, Tuesday, February 15, 2005There is a wedge product notation that can facilitate the computation of matrix Jacobians. In point offact, it is never needed at all. The reader who c an follow the deriva tion of the Jacobian in Handout #2 iswell equipped to never use wedge products. The notation also express e s the concept of volume on curvedsurfaces.For advanced reader s who truly wish to understand exterior products from a full mathematical viewpoint,this handout contains the important details. We took some pains to write a reada ble account of we dgeproducts at the cost of straying way beyond the needs of random matrix theory.The steps to understanding how to use wedge products for practical calculations are very straightforward.1. Le arn how to wedge two quantities together. This is usually understood instantly.2. Recognize that the wedge product is a forma lism for computing determinants. This is understoodinstantly as well, and at this point many readers wonder what else is needed.3. Practice wedging the order n2or mn or some such entries of a matrix together. This requires masteringthree good examples.4. Le arn the ma thematical interpretation of the wedge product of such quantities as the lower triangularpart of QTdQ and how this rela tes to the Jacobian for QR or the symmetric eigendecomposition. Weprovide a thorough explanation.1 The Mechanics of WedgingThe algebra of wedge products is very easy to master. We will wedge together differentials as illustra ted inthis example:(2dx + x2dy + 5dw + 2dz) ∧ (ydx − xdy) =(−2x − x2y)dx ∧ dy + 5y(dw ∧ dx) − 5x(dw ∧ dy) − 2y(dx ∧ dz) + 2x(dy ∧ dz)(1)Formally the wedge product ac ts like multiplication except that it follows the anticommutative law(du ∧ dv ) = (−dv ∧ du)Generally(pdu + qdv) ∧ (rdu + sdv) = (ps − qr)(du ∧ dv)It therefore follows thatdu ∧ du = 0 .In generalXfi(x) dxi∧Xgj(x) dxj=Xi<j(fi(x)gj(x) − fj(x)gi(x)) dxi∧ dxj=Xi<jfi(x) gi(x)fj(x) gj(x)dxi∧ dxj.1We can wedge together more than two differe ntials. For example2dx ∧ (3dx + 5dy) ∧ 7(dx + dy + dz) = 70dx ∧ dy ∧ dz (2)Let us write down concretely the wedge product.Rewriting the two ex amples in (1) and (2) in matrix notation, we haveF =2 yx2−x5 02 0and F =2 3 70 5 70 0 7In the first ca se, we have co mputed all 2 × 2 subdeterminants and in the second case we compute the one3 × 3 determinant. In general, if F ∈ Rn,p, we compute allnpsubdeterminants of size p.If F (x) ∈ Rn,pand dx is the vector (dx1, dx2, . . . , dxn)T, then we can wedge toge ther the elements ofF (x)Tdx. The result isp^i=1(F (x)Tdx)i=Xi1<i2<...<ipFi1i2. . . ip1 2 . . . pdxi1∧ · · · ∧ dxip,where Fi1i2. . . ip1 2 . . . pdenotes the subdeterminant of F obtained by taking rows i1, i2, . . . , ipandall p columns. In almost MATLAB notation, Fi1i2. . . ip1 2 . . . p= det(F [i1i2. . . ip, :]). In s impleEnglish, if we wedge together p differentials which we can store in the columns of F , then the wedge productcomputes all p × p subdeterminants.We use the no tation(F (x)Tdx)∧≡p^i=1(F (x)Tdx)i,i.e., “( )∧” denotes wedge together all the elements of the vector inside the parentheses.A notational note: We are inventing the “( )∧” notation. Books such as [1] use only parenthesesto indicate the wedge product of the components inside. Unfortunately, we have found that parenthesisnotation is ambiguous especially when we want to wedge together the elements of a complicated e xpression.Once we decided on placing the wedge symbol “∧” somewhere, we experimented with upp e r/lower left andupper/lower right, finally settling on the upp e r right notation as above.We will extend the “( )∧” notation from vectors to matrices of differentials. We will only wedgetogether the independent elements. For example(dM)∧=^1≤i≤m1≤j≤ndMijM ∈ Rmn.We use subscripts to indicate which elements to wedge over. For example (dS)∧i≥j=Vi≤jdSij. Whenunnecessary as in the cases below, we omit the subscripts.(dS)∧=V1≤i≤j≤ndSijS ∈ Rnnsymmetric(dA)∧=V1≤i<j≤ndAijA ∈ Rnnantisymmetric(dΛ)∧=V1≤i≤ndΛiiΛ ∈ Rnndiagonal(dU)∧=V1≤i≤j≤ndUijU ∈ Rnnupper triangular(dU)∧=V1≤i<j≤ndUijU ∈ Rnnstrictly upper triangularetc.Definitional note: We have not specified the order. Therefor e the wedge product of elements of a matrix isdefined up to “+” or “−” s ign.Notational note: We write (dM1)∧(dM2)∧for (dM1)∧∧ (dM2)∧.22 Jacobians with wedge products2.1 Wedge Notation not useful (Y = X2)Consider the 2 × 2 case of Y = X2as in Example 1 of Handout #2.With X =p qr sdY = XdX + dXX =p qr sdp dqdr ds+dp dqdr dsp qr s.ThusdY11= 2pdp + qdr + rdqdY21= rdp + (p + s)dr + rdsdY12= qdp + (p + s)dq + qdsdY22= qdr + rdq + 2sds .(3)If we wedge together all of the elements of Y , we obtain the determinant of the Jacobian:(dY )∧= 4(p + s)2(sp − qr) .The reader should compare the notation with that from the previous chapter:J =∂p ∂r ∂q ∂s2prq0qp + s0qr0p + sr0rq2s∂Y11∂Y21∂Y12∂Y22We conclude that for these examples the notation is of little value.2.2 Square Matrix = Antisymmetric + Upper TriangularGiven any matrix M ∈ Rn,n, letA = lower(M ) − lower(M)T,where lower(M) is the strictly lower triangular part of M. Further letR = upper(M) + lower(M)T,where upper(M ) includes the diagona l. We have the decompositionM=A+R= (antisymmetric) + (upper triangular) .ThereforedM =dr11dr12− da21dr13− da31· · · dr1n− dan1da21dr22dr23− da32dr2n− dan2da31da32dr33dr3n− dan3............dan1dan2dan3drnn. (4)We can easily see that the −daijplay no role in the wedge product:(dM)∧= (lower (dA) + dR)∧= (dA)∧(dR)∧.We feel this notation is already somewhat more mechanical than what was needed in the last chapter.Without wedges, we would have concentrated on the n(n + 1)/2 parameters in R and the n(n − 1)/2 inlower(A) and would have written a blo ck two by two matrix as we did in Handout #2.3The rea der sho uld think carefully about the following s e ntence: The realization that the upper daji’splay no role in (dM )∧, is equivalent to the r e alization that the matr ix X12plays no role in the deter minantof the 2 × 2