In this note a uniform transparent presentation of the scalar Haffian will be given. Some well-known results will be generalized. A link will be established between the scalar Haffian and the derivative matrix as developed by Magnus and Neudecker

Neudecker, Heinz

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QÜESTIIÓ, vol. 24, 2, p. 243-249, 2000A NOTE ON THE SCALAR HAFFIANHEINZ NEUDECKERCesaro, SchagenDepartment of EconomicsUniversity of Amsterdam∗In this note a uniform transparent presentation of the scalar H ffian wil begiven. Some well-known results will be generalized.A link will be established between the scalar Haffian and the derivativematrix as developed by Magnus and Neudecker.Keywords: Magnus-Neudecker derivative matrix, matrix vectorization,Kronecker product, duplication matrix, commutation matrixAMS Classification (MSC 2000):primary 62F10, secondary 62C99*Roetersstraat 11, 1018 WB Amsterdam, The Netherlands, e-mail:heinz@fee.uva.nl–Received July 1999.–Accepted June 2000.2431. INTRODUCTIONHaff (1977, 1979a, 1979b, 1980) introduced a scalar functiobased on the derivativesof the elements of asquare matrix functionF(X) with respect to the elements of asymmetric argument matrixX. We shall name it the scalar Haffian. It was used by Haffin various applications in multivariate statistical analysis. Several authors, among othersKonno (1988, 1991), Leung (1994) and Leung & Ng (1998) made use of it later. Theexposition and notation vary over authors and time, the derivations tend to be obscureand sometimes unnecessarily complicated.In this note we shall attempt to give a uniform transparent presentation of the scalarHaffian, and generalize some of the well-known results.Basic is a differentiablesquarematrix functionF(X), shortlyF , which depends on asymmetric matrixX. Both matrices are of the same dimension. A strategic rôle is beingplayed by a square matrix∇ = (di j ) of differential operatorsdi j := 12(1+δi j )∂∂xi j, whereδi j is the Kronecker delta(δii = 1,δi j = 0 for i 6= j).In the work mentioned earlier the symbolD is used instead of∇. We prefer∇, becauseDwill denote the so-called duplication matrix which will be ext nsively used. The matrix∇ is being applied toF and ultimately produces the scalar Haffian tr∇F , where«tr»stands for the trace operator. Haff usesD∗F(1/2) to denote this function, withF(1/2) :=1/2(F −Fd), Fd being the diagonal matrix obtained from the diagonal ofF . The scalarHaffian tr∇F will be studied in this note. It will be related to the derivati e matrix ∂ f∂x′as developed by Magnus and Neudecker (1999).In the exposition frequent use will be made of matrix vectorization, Kronecker products,the duplication matrixD and the commutation matrixK. For these concepts and someof their properties see Magnus and Neudecker (1979, 1980, 1999).2. THE SCALAR HAFFIANConsider a differentiablesquare matrix functionF(X) with symmetric matrix argu-mentX, both of dimensionm. The application of∇ = (di j ), a (square) matrix of diffe-rential operatorsdi j := 1/2(1+δi j ) ∂∂xi j to F yields∇F from which follows tr∇F , thescalar Haffian.Clearly tr∇F = ∑i jdi j f ji = ∑idii fii + ∑j 6=idi j f ji= ∑i∂ fii∂xii+12∑j 6=i∂ f ji∂xi j= ∑i∂ fii∂xii+12∑j<i∂( fi j + f ji )∂xi j244Clearly = ∑i∂gii∂xii+ ∑j<i∂gi j∂xi j= tr∂g∂x′wheregii := fii , gi j := 1/2( fi j + f ji )g := (g11· · ·gm1g22· · ·gm2 · · ·gmm)′x := (x11· · ·xm1x22· · ·xm2 · · ·xmm)′.( j < i)The expression∂g∂x′ is the Magnus-Neudecker derivative for the vector functiong(x),with x := v(X), g := v(G) andG := 1/2(F +F ′).We have thus established the identity(1) tr∇F = tr∂g∂x′for the scalar Haffian tr∇F and the Magnus-Neudecker derivative matrix∂g∂x′ .Mind that∇F 6= ∂g∂x′ ! In fact ∇F is another useful concept also developed and appliedby Haff.See Haff (1981, 1982). Obviously the scalar Haffian can then also be obtained from∇F . We shall name∇F the matrix Haffian. It will be examined in another paper.An attractive alternative expression for the scalar Haffianis tr ∂v(F+F′)∂v′(X) which showsimmediately that(2) tr∇F ′ = tr∇F.WhenF is symmetric(3) tr∇F = tr∂ f∂x′where f := ( f11· · · fm1 f22· · · fm2 · · · fmm)′ = v(F).ProofIn the creation of (1) we now havefi j = f ji , , henceg = f .¤3. A GENERAL RESULTInstead of deriving umpteen specific scalar Haffians we shallestablish a general resultfrom which other specific results can be derived.245TheoremFor symmetric X and square constant matrices P and Qtr∇PXQ′ = 1/2(trP) trQ+1/2trPQ.ProofTakeF := PXQ′. AgainG := 1/2(F +F ′). ThendvecG = 1/2(dvecF +dvecF ′) = 1/2(Im2 +Kmm)dvecF,anddg = 1/2D+m(Im2 +Kmm)dvecF = D+mdvecF= D+mvecP(dX)Q′ = D+m(Q⊗P)dvecX= D+m(Q⊗P)Dmdx.Hence∂g∂x′= D+m(Q⊗P)Dm.Therefromtr∇F = trD+m(Q⊗P)Dm = trDmD+m(Q⊗P)= 1/2tr(Im2 +Kmm)(Q⊗P) = 1/2tr(Q⊗P)+1/2trKmm(Q⊗P)= 1/2(trP)trQ+1/2trQP= 1/2(trP)trQ+1/2trPQ.We used various results from Magnus and Neudecker (1999, pp.30, 47 and 49) andMagnus and Neudecker (1979, Theorem 3.1, xiv).¤CorollaryFor any function F= F(X) such that dF= P(dX)Q′, the scalar Haffian istr∇F = 1/2(trP)trQ+1/2trPQ.With the help of this corollary we can now derive scalar Haffians in practice.2464. VARIOUS SCALAR HAFFIANS(ii) tr∇PX−1Q′ = −1/2(trPX−1)trQX−1−1/2trPX−1QX−1.ProofNow F := PX−1Q′ anddF = P(dX−1)Q′ = −PX−1(dX)X−1Q′.Replacing thenP by−PX−1 andQ′ by X−1Q′ in the Corollary, one immediately obtainstr∇PX−1Q′ = −1/2(trPX−1)trQX−1−1/2trPX−1QX−1.¤(ii ) tr∇PXQXR′ = 1/2(trP)trRXQ′+1/2trPRXQ′+1/2trPXQR+1/2(trR)trPXQ.ProofAs F := PXQXR′ anddF = P(dX)QXR′+PXQ(dX)R′ we have to make the followingsubstitutions:{P PQ′ QXR′and{P PXQQ′ R′This then leads to the scalar Haffiantr∇PXQR′ = 1/2(trP)trRXQ′ +1/2trPRXQ′ +1/2(trR)trPXQ+1/2trPXQR.¤(iii ) tr∇PX−2Q′ = = −1/2(trPX−1) trQX−2−1/2(trPX−2) trQX−1− 1/2trPX−1QX−2−1/2trPX−2QX−1.ProofIn this caseF := PX−2Q′ anddF = P(dX−2)Q′ = P(dX−1)X−1Q′ +PX−1(dX−1)Q′= −PX−1(dX)X−2Q′−PX−2(dX)X−1Q′.We shall make the following substitutions{P −PX−1Q′ X−2Q′and{P −PX−2Q′ X−1Q′and get the above given result. ¤247(iv) tr∇PX3Q′ = 1/2(trP)trQX2 +1/2trPQX2 +1/2(trPX)trQX+ 1/2trPXQX+1/2trPX2Q+1/2(trQ)trPX2.ProofNow F := PX3Q′, hencedF = P(dX)X2Q′ +PX(dX)XQ′ +PX2(dX)Q′,which leads to the substitutions{P PQ′ X2Q′,{P PXQ′ XQ′and{P PX2Q′ Q′Hence the scalar Haffian obtains.¤NOTES1. Haff (1979a, 1980), Konno (1988) and Leung (1994) considere tr∇XQ′ and tr∇PX,with occasionally positive definiteQ andX.2. Clearly the Theorem also holds for symmetricP,Q andPXQ′.3. tr∇X−1Q′ was derived by Haff (1979a), tr∇X−1 was given by Haff (1980) for posi-tive definiteX.4. tr∇X2Q′ was derived by Haff (1979a), the identical tr∇QX2 was found by Konno(1991). In fact these are special cases of (ii ).5. Konno (1988) gave tr∇XQX, with positive definite X. Leung (1994) and Leung &Ng (1998) considered tr∇XQX with symmetric, even positive definiteQ.6. Haff (1980) presented tr∇X−2 for positive definiteX.7. Konno (1991), gave tr∇X3 for positive definiteX.ACKNOWLEDGMENTThanks to the advice of the referee several improvements in the exposition could berealized.248REFERENCESHaff, L.R. (1977).«Minimax estimators for a multinormal precision matrix». J. Mul-tivar. Anal., 7, 374-85.Haff, L.R. (1979a).«An identity for the Wishart distribution with applications». J. Mul-tivar. Anal., 9, 531-44.Haff, L.R. (1979b).«Estimation of the inverse covariance matrix: random mixtures ofthe inverse Wishart matrix and the identity». Ann. Statist., 7, 1264-76.Haff, L.R. (1980).«Empirical Bayes estimation of the multivariate normal covariancematrix». Ann. Statist., 8, 586-97.Haff, L.R. (1981).«Further identities for the Wishart distribution with applications inregression». Canad. J. Statist., 9, 215-24.Haff, L.R. (1982).«Identities for the inverse Wishart distribution with computationalresults in linear and quadratic discrimination». Sankhỹa B, 44, 245-58.Konno, Y. (1988).«Exact moments of the multivariate F and beta distributions». J. Ja-pan Statist. Soc., 18, 123-30.Konno, Y. (1991).«A note on estimating eigenvalues of (the) scale matrix of themul-tivariate F-distribution». Ann. Inst. Statist. Math., 43, 157-65.Leung, P.L. (1994).«An identity for the noncentral Wishart distribution with applica-tion». J. Multivar. Anal., 48, 107-14.Leung, P.L. and Ng, F.Y. (1998).Improved estimation of parameter matrices in a one-sample and two-sample problem. Department of Statistics, The Chinese Universityof Hong Kong.Magnus, J.R. and Neudecker, H. (1979).«The commutation matrix: some properties andapplications». Ann. Statist., 7, 381-94.Magnus, J.R. and Neudecker, H. (1980).«The elimination matrix: some lemmas andapplications». SIAM J. Alg. and Discr. Meth., 1, 422-49.Magnus, J.R. and Neudecker, H. (1999).Matrix Differential Calculus with Applicationsin Statistics and Econometrics, revised edition. John Wiley, Chichester.249

A note on the scalar Haffian

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A note on the scalar Haffian

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