Proceedngs of the 9th WSEAS Internatonal Conference on Aled Mathematcs, Istanbul, Turkey, May 27-29, 2006 (489-494) The Matrx Analogs of Frey s Extenson of Mnkowsk Inequalty and of Frey s Extenson of Brunn-Mnkowsk Inequalty PORAMATE (TOM) PRANAYANUNTANA, JOHN GORDON 2 Faculty of Engneerng, Kng Mongkut s Insttute of Technology Ladkrabang (KMITL) 3 Moo 2 Chalongkrung Rd., Ladkrabang, Bangkok 0520 THAILAND Tel: (662) 326-422, Fax: (662) 326-4225 korama@kmtl.ac.th, ranaya@oly.edu 2 Queensborough Communty College Baysde, NY 364 USA Tel: (78) 63-636 jgordon@qcc.cuny.edu Abstract: The Brunn-Mnkowsk theory s a central art of convex geometry. At ts foundaton les the Mnkowsk addton of convex bodes whch led to the defnton of mxed volume of convex bodes and to varous notons and nequaltes n convex geometry. Its orgns were n Mnkowsk s jonng hs noton of mxed volumes wth the Brunn-Mnkowsk nequalty, whch dated back to 887. Snce then t has led to a seres of other nequaltes n convex geometry. The exstence s very useful and wdely used n mathematcal and engneerng alcatons. Our urose of ths seres was to develo an equvalent seres of nequaltes for ostve defnte symmetrc matrces. The major theorems resented here are the matrx analogs of Frey s extenson of Mnkowsk nequalty and of Frey s extenson of Brunn-Mnkowsk nequalty. Key-Words: Aleksandrov nequalty, Quermassntegral, Mxed Quermassntegral, Matrx Frey -Sum, Mxed -Quermassntegral, matrx analog of Frey s Extenson of Brunn-Mnkowsk nequalty, matrx analog of Frey s Extenson of Mnkowsk nequalty. Introducton The Brunn-Mnkowsk theory s the heart of quanttatve convexty. Its orgns were n Mnkowsk s jonng hs noton of mxed volumes wth the Brunn- Mnkowsk nequalty, whch dated back to 887. Snce then t has led to a seres of other nequaltes n convex geometry. The exstence s very useful and wdely used n mathematcal and engneerng alcatons. Our urose of ths seres was to develo an equvalent seres of nequaltes for ostve defnte symmetrc matrces. Some of these new matrx analogous notons and nequaltes can be found n [9], among them the mortant ones are the matrx verson of Blaschke summaton [9], matrx verson of arallel summaton [3, 9, 2], matrx verson of arallel Blaschke summaton [9], and the matrx analog of the Kneser- Süss nequalty [9]. In ths aer, our goal here s to develo two new matrx analogs of the nequaltes n convex geometry, namely, the matrx analog of Frey s Extenson of Mnkowsk nequalty and the matrx analog of Frey s Extenson of Brunn-Mnkowsk nequalty. 2. Materals and Methods 2. Matrx Quermassntegrals, Matrx Mxed Quermassntegrals, Frey s Sum and Matrx Mxed -Quermassntegrals Let A be an n n matrx wth real entres and we denote ts jth entry by (A) j. We wll denote by A or D(A) or det(a) the determnant of A. From ths ont on, unless stated otherwse, we assume that all matrces are symmetrc. Let Mn s denote the set of all n n ostve defnte symmetrc matrces. For A,..., A n Mn, s let D(A,..., A n ) denote the mxed determnant [9] of A,..., A n.
Proceedngs of the 9th WSEAS Internatonal Conference on Aled Mathematcs, Istanbul, Turkey, May 27-29, 2006 (489-494) Remark (Mxed Determnant [9]). A mxed determnant D(A, A 2,..., A n ) of n n matrces A, A 2,..., A n can be regarded as the arthmetc mean of the determnants of all ossble matrces whch have exactly one row from the corresondng rows of A, A 2,..., A n. For examle, for any A, B Mn s, the mxed determnant D(A, n ; B, ) s a b D(A,..., A, B) =. a 2 + + }{{} n n a n.. () We can easly rove that nd (A, B) = nd(a, n D(A + B) D(A) ; B, ) = lm = CA B, 0 + where A B := (A) j (B) j and CA s the cofactor matrx of A, the transose of the classcal adjont,j matrx of A. Defnton 2 (Matrx Quermassntegrals). For A M s n, let W 0 (A), W (A),..., W n (A) denote the matrx Quermassntegrals of A defned by b n a n W (A) = D(A, n ; I, ), (2) the mxed determnant wth n coes of A and coes of I. Thus, W 0 (A) = D(A), the determnant of A, W (A) = D(A, n ; I, ) = D (A, I) = CA I = n n tr(ca) = n D(A) tr(a ) = n n D(A) λ, = and W n (A) = D(A, ; I, n ) = D(I, n ; A, ) = D (I, A) = CI A = I A n n = n tr A = n λ the mean of egenvalues of A, where n = λ,..., λ n are the egenvalues of A Mn s. Defnton 3 (Matrx Mxed Quermassntegrals). The matrx mxed Quermassntegrals W 0 (A, B), W (A, B),..., W n (A, B) of A, B M s n are defned by (n )W (A, B) = lm 0 + W (A + B) W (A). Snce W (ca) = c W (A), t follows that W (A, A) = W (A), for all. Snce the Quermassntegral W n s Mnkowsk lnear, t follows that W n (A, B) = W n (B), A. The mxed Quermassntegral W 0 (A, B) wll usually be wrtten as D (A, B). The fundamental nequalty for mxed Quermassntegrals n convexty theory states that: For K, L K n and 0 < n, W (K, L) W (K) W (L), (3) wth equalty f and only f K and L are homothetc. Good general references for ths materal are Busemann [2] and Lechtweß [8]. Its matrx verson states as follows: For A, B Mn s and 0 < n, W (A, B) W (A) W (B), (4) wth equalty f and only f A and B are scalar multle of each other. Ths matrx nequalty can be roved easly usng Aleksandrov nequalty [, ], whch s the matrx verson of the Aleksandrov-Fenchel nequalty for mxed volumes. Defnton 4 (Matrx Frey Summaton). Let A, B M s n. Then the matrx Frey sum of A and B, denoted A + B, s defned by A + B := (A + B ) /. (5) The commutatvty and the assocatvty of + are obvous. Mxed Quermassntegrals are, of course, the frst varaton of the ordnary Quermassntegrals, wth resect to Mnkowsk addton. The frst varaton of the ordnary Quermassntegrals wth resect to Frey addton s as follows: Defnton 5 (Matrx Mxed -Quermassntegral). Defne the mxed -Quermassntegrals W,0 (A, B), W, (A, B),..., W,n (A, B) as the frst varaton of the ordnary Quermassntegrals, wth resect to Frey addton: For A, B M s n, and, 0 n, the mxed - Quermassntegrals of A, B, denoted W, (A, B), s defned by n W, (A, B) = lm 0 + W (A + B) W (A) where B = / B exresses the relaton between Frey scalar multlcaton ( ) and Mnkowsk scalar multlcaton. Of course for =, the mxed -Quermassntegral W, (A, B) s just W (A, B). Obvously, W, (A, A) = W (A), for all.
Proceedngs of the 9th WSEAS Internatonal Conference on Aled Mathematcs, Istanbul, Turkey, May 27-29, 2006 (489-494) Lutwak [7] states that for these new mxed Quermassntegrals, there s an extenson of nequalty (3): If K, L K n 0, 0 n, and >, then W, (K, L) W (K)W (L), (6) wth equalty f and only f K and L are dlates. We wll rove ts matrx verson: If A, B M s n, 0 n, and >, then W, (A, B) W (A)W (B), (7) wth equalty f and only f B = c A, c > 0 n Theorem 8. Theorem 6. (a) For A, B Mn s and α, β > 0, W, (αa, βb) = α β W, (A, B), and when = and β =, W, (αa, B) = W, (A, B) for all α > 0. (b) For all Q, A, B Mn, s W, (Q, A + B) = W, (Q, A) + W, (Q, B). Ths result shows that the mxed -Quermassntegral s lnear, wth resect to Frey -sums, n ts second argument. 2.2 Varatonal Characterzatons of Egenvalues of Symmetrc Matrces For a general matrx A M n, about the only characterzaton of the egenvalues s the fact that they are the roots of the characterstc equaton A (t) = 0. For symmetrc matrces, however, the egenvalues can be characterzed as the solutons of a seres of otmzaton roblems. Snce the egenvalues of a symmetrc matrx A M n are real, we shall adot the conventon that they are labeled accordng to ncreasng (non-decreasng) sze: λ mn = λ λ 2 λ n λ n = λ. (8) The smallest and largest egenvalues are easly characterzed as the solutons to a constraned mnmum and mum roblem. The characterzaton theorem bears the names of two Brtsh hyscsts, and the exresson xt Ax s known as a Raylegh-Rtz rato. Theorem 7 (Raylegh-Rtz [4]). Let A M n be symmetrc, and let the egenvalues of A be ordered as n (8). Then λ λ n x R n, λ = λ n = x 0 = = xt Ax, λ mn = λ = mn x 0 = mn = xt Ax. The Raylegh-Rtz theorem rovdes a varatonal characterzaton of the largest and smallest egenvalues of a symmetrc matrx A, but what about the rest of the egenvalues? Ths queston s answered n the followng Courant-Fscher mn- theorem. Theorem 8 (Courant-Fscher [4]). Let A M n be a symmetrc matrx wth egenvalues λ λ 2 λ n, and let k be a gven nteger wth k n. Then and mn w,w 2,...,w n k R n x 0, x R n x w,w 2,...,w n k mn w,w 2,...,w k R n x 0, x R n x w,w 2,...,w k = λ k (9) = λ k. (0) Remark: If k = n n (9) or k = n (0), we agree to omt the outer otmzaton, as the set over whch the otmzaton takes lace s emty. In the two cases the assertons reduce to the Raylegh-Rtz theorem (Theorem 7). 2.3 Some Alcatons of the Varatonal Characterzatons Among the many mortant alcatons of the Courant-Fscher theorem, one of the smlest s to the roblem of comarng the egenvalues of A + B wth those of A. We denote the egenvalues of a matrx A by λ (A). Theorem 9 (Weyl [4]). Let A, B M n be symmetrc and let the egenvalues λ (A), λ (B), and λ (A+B) be arranged n ncreasng order (8). For each k =, 2,..., n we have λ k (A) + λ (B) λ k (A + B) λ k (A) + λ n (B).() Proof. bound For any nonzero x R n we have the λ (B) xt Bx λ n(b) and hence for any k =, 2,..., n we have λ k (A + B) = mn w,...,w n k R n x 0 x w,...,w n k = mn w,...,w n k R n x 0 x w,...,w n k mn w,...,w n k R n x 0 x w,...,w n k = λ k (A) + λ (B). x T (A + B)x [ + xt Bx ] [ x T ] Ax + λ (B)
Proceedngs of the 9th WSEAS Internatonal Conference on Aled Mathematcs, Istanbul, Turkey, May 27-29, 2006 (489-494) A smlar argument establshes the uer bound as well. matrx A B = [a j B] m,j= M mn. Weyl s theorem gves two-sded bounds for the egenvalues of A + B for any symmetrc matrces A and B. Further refnements can be obtaned by restrctng B to have a secal form for examle, ostve defnte, rank, rank k, or boderng matrx. Recall that a matrx B M n such that x T Bx 0 for all x R n s sad to be ostve semdefnte; an equvalent condton s that B s symmetrc and have all egenvalues nonnegatve. For symmetrc matrces A, B we wrte B A or A B to mean that A B s ostve semdefnte. In artcular, A 0 ndcates that A s ostve semdefnte. Ths s known as the Löwner artal order. If A s ostve defnte, that s, ostve semdefnte and nvertble, we wrte A > 0. The followng result, an mmedate corollary of Weyl s theorem known as the monotoncty theorem, says that all the egenvalues of a symmetrc matrx ncrease f a ostve semdefnte matrx s added to t. Corollary 0 ([4]). Let A, B M n be symmetrc. Assume that B s ostve semdefnte and that the egenvalues of A and A + B are arranged n ncreasng order (8). Then λ k (A) λ k (A + B), k =, 2,..., n. Corollary ([4]). If A, B M n are ostve defnte symmetrc, then f A B, then det A det B and tr A tr B; and more generally, f A B, then λ k (A) λ k (B) for all k =, 2,..., n f the resectve egenvalues of A and B are arranged n the same (ncreasng or decreasng) order. 2.4 Some Useful Results from Löwner Partal Order A ma Φ : M m M n s called ostve f t mas ostve semdefnte matrces to ostve semdefnte matrces: A 0 Φ(A) 0. Φ s called untal f Φ(I m ) = I n. Gven A = [a j ] M m, B = [b j ] M n, then the rght Kronecker (or drect, or tensor) roduct of A and B, wrtten A B, s defned to be the arttoned Gven A = [a j ] M m, B = [b j ] M n, the Hadamard roduct of A and B s defned as the entrywse roduct: A B [a j b j ] M n. We denote by A[α] the rncal submatrx of A ndexed by α. The followng observaton s very useful. Lemma 2 ([5, 2]). For any A, B M n, A B = (A B)[α] where α = {, n + 2, 2n + 3,..., n 2}. Consequently there s a untal ostve lnear ma Φ from M n 2 to M n such that Φ(A B) = A B for all A, B M n. As an llustraton of the usefulness of ths lemma, consder the followng reasonng: If A, B 0, then evdently A B 0. Snce A B s a rncal submatrx of A B, A B 0. Smlarly A B > 0 for the case when both A and B are ostve defnte. In other words, the Hadamard roduct of ostve semdefnte (defnte) matrces s ostve semdefnte (defnte). Ths mortant fact s known as the Schur roduct theorem. Let P n be the set of ostve semdefnte matrces n M n. A ma Ψ : P n P n P m s called jontly concave f Ψ(λA + ( λ)b, λc + ( λ)d) λψ(a, C) + ( λ)ψ(b, D). for all A, B, C, D 0 and 0 < λ <. For A, B > 0 Mn s the arallel sum of A and B s defned as A +B := ( A + B ). We have the extremal reresentaton for the arallel sum of A, B P n : { [ ] } A + B A A +B = X 0 : 0 A A X where the mum s wth resect to the Löwner artal order. From ths extremal reresentaton we can show that the ma (A, B) A +B s jontly concave [0, 2]. Lemma 3 ([0, 2]). For 0 < r < the ma (A, B) A r B r s jontly concave n A, B 0.
Proceedngs of the 9th WSEAS Internatonal Conference on Aled Mathematcs, Istanbul, Turkey, May 27-29, 2006 (489-494) Theorem 4. For A, B, C, D 0 and, q > wth / + /q =, A B + C D (A + C) (B + q D). Proof. Ths s just the md-ont jont concavty case λ = /2 of Lemma 3 wth r = /. 2.5 Man Results Theorem 5. For X, Y > 0 M s n and [0, ] we have ( )X + Y (( )X + Y ) / =: ( ) X + Y Proof. Ths s just Theorem 4 wth A = ( ) / X, B = ( ) /q n, C = / Y and D = /q n, where n = [] n n, that s n s the n n matrx that has all of ts entres equal to. Alyng Corollary to the nequalty n Theorem 5 and the fact that for any matrx A M n, W 0 (A) = λ λ 2 λ n, W (A) = λ λ n,..., W n 3 (A) = λ λ 2 λ 3, W n 2 (A) = λ λ 2, W n (A) = λ. We can easly see that for any A, B > 0 W (( ) A + B) W (( )A + B) (( )W (A) + W (B) ), 0 n, (2) where the last nequalty s by alyng the Aleksandrov-Fenchel nequalty. Theorem 6. If A 0, B 0 > 0 M s n wth W (A 0 ) = W (B 0 ) = then W (α A 0 + ( α) B 0 ) α [0, ], 0 n. Proof. Aly (2) to A 0, B 0 wth = α. The followng two theorems are analogs of Frey s Extenson of Brunn-Mnkowsk and Mnkowsk theorems (see Lutwak [7] for the orgnal theorems n convex geometry). Theorem 7 (Matrx Analog of Frey s Extenson of Brunn-Mnkowsk Inequalty). If A, B M s n, 0 n, >, then W (A + B) W (A) + W (B) (3) wth equalty f and only f A = c B, c > 0. Proof. We aly Theorem 6 wth A 0 = A, W (A) to obtan W ( W B 0 = α = W (B) B, W (A) W (A) + W (B) W (A) + W (B) A + W (A) + W (B) B A + B ) (W (A) + W (B) ) ) W (A (W + B ) (A) + W (B) W (A + B ) or W ((A + B ) ) that s, W (A + B) W (A) + W (B) W (A) + W (B), W (A) + W (B). The equalty art can be seen by drectly substtutng A = c B. Ths comletes the roof. Theorem 8 (Matrx Analog of Frey s Extenson of Mnkowsk Inequalty). If A, B M s n, 0 n, >, then W, (A, B) W (A)W (B) (4) wth equalty f and only f A = c B, c > 0. Proof. Theorem 7 mles W B) ( )W Then lm 0 (A) + W (( ) A + (B), and snce W (( ) A + B) W (A) = n W,(A, B) = W (A) + W (A) + n lm 0 W,(A, B) + n W (A). W (( ) A + B) W (A) n lm 0» ()/ ( )W (A) + W (B) W (A) = W (A) +» n h W (A) /()[()/ ] W (B) /() W (A) n
Proceedngs of the 9th WSEAS Internatonal Conference on Aled Mathematcs, Istanbul, Turkey, May 27-29, 2006 (489-494) = W (A) /() W (B) /(). Ths comletes the roof. Furthermore, we can also show that the nequaltes (3) and (4) are equvalent. Snce we already show that (3) mles (4), t suffces to show that (4) mles (3): Snce W, (A, B) W (A)W (B), for A, B Mn, s 0 < n, >, and W, (Q, A + B) = W, (Q, A) + W, (Q, B) then W, (Q, A + B) W (Q)[W (A) + W (B)]. We now set A + B equal to Q and use the fact that W, (Q, Q) = W (Q) to obtan» W (A + B) W (A + B) W (A) + W (B), or W whch s (3). (A + B) W (A) + W (B) 3. Acknowledgment The frst author dedcates ths aer to Assocate Professor Dr. Chandn Shah, Se 3, 959 - Ar 9, 2005, who was lke a mother to hm. We wshes to thank Professor Erwn Lutwak for some very helful and nsrng conversatons (and corresondence) on the subject of ths artcle. [6] Erwn Lutwak, Volume of Mxed Bodes, Transactons of The Amercan Mathematcal Socety, vol. 294, 2, Arl 986,. 487-500. [7] Erwn Lutwak, The Brunn-Mnkowsk-Frey Theory I: Mxed Volumes and the Mnkowsk Problem, Journal of Dfferental Geometry, 38, 993,. 3-50. [8] K. Lechtweß, Konvexe Mengen, Srnger, Berln, 980. [9] Poramate (Tom) Pranayanuntana, Patcharn Hemchote, Praboon Pantaraghong, The Matrx Analog of the Kneser-Süss Inequalty, to be aeared n WSEAS Proceedngs of MATH, TELE-INFO and SIP 06, Istanbul, Turkey, May 27-29, 2006. [0] Poramate (Tom) Pranayanuntana, John Gordon, New Matrx Inequaltes for Frey s Extenson of Mnkowsk and Brunn-Mnkowsk Inequaltes, to be aeared n WSEAS Journal. [] Rolf Schneder, Convex Bodes: The Brunn-Mnkowsk Theory, Cambrdge Unversty Press, New York, 993. [2] Xngzh Zhan, Matrx Inequaltes, Lecture Notes n Mathematcs, vol. 790, Srnger, New York, 2002. Polytechnc Unversty Brooklyn, NY The frst author thanks Professor Xngzh Zhan and Assstant Professor Franzska Berger for some very useful suggestons, also thanks Dr.Alna Stancu for her many excellent suggestons for mrovng the orgnal manuscrt. Thanks Professor Tawl Paungma, Dean of the Faculty of Engneerng, KMITL, for allocatng funds for ths research. He also thanks ther Faculty of Engneerng for the resources rovded. References: [] A. D. Aleksandrov, Zur Theore der gemschten Volumna von konvexen Körern, IV. De gemschten Dskrmnanten und de gemschten Volumna (n Russan), Mat. Sbornk N.S. 3, 938,. 227-25. [2] H. Busemann, Convex Surfaces, Interscence, New York, 958. [3] G. P. Egorychev, Mxed Dscrmnants and Parallel Addton, Sovet Math. Dokl., vol. 4, 3, 990,. 45-455. [4] Roger A. Horn and Charles R. Johnson, Matrx Analyss, Cambrdge Unversty Press, New York, 985. [5] Roger A. Horn and Charles R. Johnson, Tocs n Matrx Analyss, Cambrdge Unversty Press, New York, 99.