ON SOME MATRIX INEQUALITIES. Hyun Deok Lee. 1. Introduction Matrix inequalities play an important role in statistical mechanics([1,3,6,7]).

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1 Korean J. Math. 6 (2008), No. 4, pp ON SOME MATRIX INEQUALITIES Hyun Deok Lee Abstract. In this paper we present soe trace inequalities for positive definite atrices in statistical echanics. In order to prove the ethod of the unifor bound on the generating functional for the sei-classical odel, we use soe trace inequalities and atrix nors and properties of trace for positive definite atrices.. Introduction Matrix inequalities play an iportant role in statistical echanics([,3,6,7]). We study quantu statistical echanics for the seiclassical odel in the lattice space. In order to investigate the unifor bound on the generating functional, the sei-classical odel has been studied intensively by any authors([,3,6,7,0]). The purpose of this paper is to establish the trace inequality for ultiple product of powers of arbitrary finite positive definite atrices which is a tool to find the unifor bound on the generating functional for the sei-classical odel in quantu statistical echanics. This paper is organized as follows. In section II, We introduce soe definitions and theores which are necessary to prove our ain result and present soe properties of trace for positive definite atrices on the finite diensional Hilbert space([ 2,4, ]). In section III, we first describe soe trace inequalities, and then we will prove our ain result by applying these trace inequalities and properties of atrix nors,and induction arguent. 2. Preliinaries Let H n denote the coplex vector space of all n n Heritian Received Noveber 20, Revised Deceber 7, Matheatics Subject Classification: 82B0. Key words and phrases: positive definite atrices, atrix nors, trace. This work was supported by the Cheongju University Scholar Overseas Funds.

2 566 Hyun Deok Lee atrices, endowed with the inner product < A, B >= T r(b A), where T r( ) is the trace on the positive atrices and B is the adjoint of B, Then,this akes (H n, <, >) into a Hilbert space H([,2]). We first define an n n Heritian atrix A is said to be positive definite, denoted by, A > 0, if x Ax > 0 for all nonzero x in C n. If x Ax 0,then A is said to be positive seidefinite, denoted by, A 0. Definition 2.. ([0,2]) Let A in H n, Then, ()The trace nor of A, defined by, A = i s i(a). (2) The spectral nor of A,also denoted by, A 2 = ax {s i (A)}, where s i (A) are the singular values of A, i.e., the eigenvalues of A = (A A) 2. Theore 2.2. ([6])Let A be a positive definite atrix in H n with finite trace nor and B be a positive atrix with finite trace nor, Then, ()T r(uau ) = T r(a) for any unitary atrix U, (2)T r(ab) = T r(ba). We now restrict to the case the finite diension of H, i.e., di(h) < Theore 2.3. ([])[Cauchy-Schwartz inequality] Let A and B be positive definite atrices with finite trace nor,respectively, then, T r(a B 2 T r(a A)T r(b B). By the Cauchy-Schwartz inequality, we have Theore 2.4. ([]) Let A and B be positive definite atrices with finite trace nor, respectively, then, ()T r(ab) T r(ab {T r(a A)} 2 {T r(b B)} 2, (2)T r(ab) < T r(a)t r(b).

3 3. Trace Inequalities On Soe Matrix Inequalities 567 In order to prove our ain result, it is necessary to establish the following two leas. Lea 3.. ([4,5,9])Let A 0 and B 0 in H n, then, T r(ab) A 2 T r(b), where A 2 denotes the spectral nor or largest singular value of A. Lea 3.2. ([9, ])For any positive atrices C,D, and E in H n, then, T r(cde D 2 {T r(c C)} 2 {T r(e E)} 2. Proof. T r(cde D 2 T r(ce) By Theore 2.2 and Theore 2.4, cobining the Cauchy-Schwartz inequality, we obtain the following : T r(cde D 2 T r(ce) D 2 {T r(c C)} 2 {T r(e E)} 2. Now, we will prove our ain result using above two leas. Theore 3.3. Let B, B 2,..., B n be positive definite atrices with finite atrix nor in H n, and A be a positive definite atrix such that A n has a finite trace nor, then, for p i 0, i =, 2,..., n, with n i= p i =, T r(b A p B 2 A p2 B n A p n ( n B i 2 )T r(a). Proof. We will prove the theore by induction. For n =, it follows fro Lea 3.. For n = 2, let p 2,then, T r(b A p i= = T r(a p +p 2 2 B A p B 2 A p p 2 2

4 568 Hyun Deok Lee B 2 T r(a) 2 T r(a p p 2 2 B 2A 2p B 2 A p p 2 2 ) 2, by Lea 3.2. Continuing in this process, after n-steps,we obtain T r(b A p 2 B 2 B n T r(a) n T r(b2a p for soe p 0, p 2 0 with p + p 2 =. Since one of p i s is less than 2, we have T r(b 2A p 2 n B n A 2 2 n T r(a 2 ) By taking n, we proved the theore for n = 2. We now assue that the theore holds for n. We will show that the theore holds for n =. Since p + p p =, there exists j in N such that 2 n. 2 n p j + p j+ (od ) p j+[ 2 ] (od ) < 2 p j + p j+ (od ) p j++[ 2 ] (od ) 2, where [ 2 ] is the largest integer which is not greater than 2. Using the cyclic property of the trace : T r(ab) = T r(ba) and rearranging B j and p j, we ay assue that p + p p [ 2 ] < 2 p + p p [ 2 ]+ 2 () Let p = 2 [ 2 ] i= p i and let = [ 2 ] +, then, p i + p ) = 2( i=

5 On Soe Matrix Inequalities 569 and 2( p i + (p p )) = (2) i= + T r(b A p B 2 A p2 B A p T r(a p p B + Ap + B A p B A p B A p B 2 T r(a p B Ap B...B 2A 2p...B A p ) 2 T r(a p p B +...B A 2p B A p...b + Ap p ) 2 (3) If is odd, then 2( ) = and 2( ( + )) =. Thus, for odd integer, the theore follows fro (2) and the induction hypothesis. Next, let be even, then 2( ) = and 2( ( +)) = 2. So, by the induction hypothesis, (3) is bounded by B 2 j= + T r(a) 2 T r(b Ap B...B 2A 2p B 2...B A 2p ) 2 (4) Notice that by () either p j < 2 and or else p j + 2p 2 (5)

6 570 Hyun Deok Lee p j < 2 and p j + 2p 2 (6) In either case, we use the ethod to obtain (3) and (4). ( T r(b Ap B B 2 A 2p...B A 2p ) B j 2 T r(a) 2 T r(a p B B 2A 2p B2 A p 2 B A p ) for soe p,..., p with 2(p p ) = and such that one of (5) and (6) holds for p,..., p After n-steps of the above continuing process, we conclude that T r(b A p...b A p B 2 ( j= + B j 2 )T r(a) n ( B j 2 ) n T r(a q B B 2A 2q B 2 B A q 2 n (7) for soe q,..., q with 2(q q ) = Since one of q i s is less than, the trace in (7) is bounded by ( B j 2 n 2 ) A ( ) 2 n T r(a ) 2 n. Hence, The theore follows fro (7). This copletes the proof of the theore.

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