A generalization of a trace inequality for positive definite matrices

Transcription

1 A generalzaton of a trace nequalty for postve defnte matrces Elena Veronca Belmega, Marc Jungers, Samson Lasaulce To cte ths verson: Elena Veronca Belmega, Marc Jungers, Samson Lasaulce. A generalzaton of a trace nequalty for postve defnte matrces. Australan Journal of Mathematcal Analyss and Applcatons, Austral Internet Publshng, 0, 7 (), pp.art. 6, -5. <hal > HAL Id: hal Submtted on 5 Jan 00 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 A GENERALIZATION OF A TRACE INEQUALITY FOR POSITIVE DEFINITE MATRICES. ELENA VERONICA BELMEGA, MARC JUNGERS, AND SAMSON LASAULCE Abstract. In ths note we generalze the trace nequalty derved by [] to the case where the number of terms of the sum (denoted by ) s arbtrary. More precsely we ( ) ( ) prove that T = Tr k B k ) B l A l 0 for any set of k= postve defnte matrces. l= l=. Introducton Trace nequaltes are useful n many areas lke multple nput multple output (MIMO) systems n control theory and communcatons. Provng the trace nequalty under nvestgaton n ths note s a suffcent condton to ensure the unqueness of a Nash equlbrum n certan MIMO communcatons game [] where Rosen s dagonally strct concavty condton [3] s vald. The consdered nequalty has been proven by e.g., [4] for = and by [] for =. Here we generalze t to N. The man result of ths note s as follows. Theorem.. Let N. Assume that (): A = A H 0, B = B H 0; (): k,...,, A k = A H k 0 and B k = B H k 0. Then, we have that ( ) ( ) (.) T Tr k B k ) B l A l 0. k= l=. Auxlary Results In order to prove Theorem., we wll use two auxlary lemmas whch are stated here for the sake of clarty. Lemma.. [] Let A, B be two postve defnte matrces, C, D, two postve semdefnte matrces whereas X s only assumed to be Hermtan. Then (.) Tr XA XB Tr X C) X(B D) 0. The proof s gven n []. Lemma.. Let A, B be two postve defnte matrces, C, D, two postve sem-defnte matrces. Then Tr B)(B D) (.) (C D) C) = Tr (C D)(B D) B) C) R. 000 Mathematcs Subject Classfcaton. 5A45. ey words and phrases. Trace nequalty, postve defnte matrces, postve semdefnte matrces. l=

3 ELENA VERONICA BELMEGA, MARC JUNGERS, AND SAMSON LASAULCE Proof. To prove the desred result, let us defne E by (.3) E = Tr (C D) [ (B D) C) ] and wrte t n two dfferent ways. (.4) E = Tr (C D)(B D) [A C B D] C) = Tr (C D)(B D) (C D) C) Tr (C D)(B D) B) C). (.5) E = Tr (C D) C) [A C B D](B D) = Tr (C D) C) (C D)(B D) Tr (C D) C) B)(B D). By usng the commutaton property of the trace and the two expressons of E, we fnd the desred result. The only thng whch needs to be proven s that E s real. For ths purpose, observe that f we denote by M = (C D)(B D) B) C) then M H = C) B)(B D) (C D) and from the result just proven we obtan that Tr(M H ) = Tr(M) and thus we have that Tr(M) R. Defne X k = 3. Proof of Theorem. A, Y k = B, for all k whch are both postve defnte = = matrces. Notce that T can be re-wrtten recursvely as follows: T = Tr B )Y B )X (3.) T = T Tr B )Y B )X Tr B )Y (X Y )X We proceed n two steps. Frst, we fnd a lower bound for T and then we prove that ths bound s postve. Frst, let us prove that, for all : (3.) T = Tr B )Y B )X Y )Y (X Y )X To ths end we proceed by nducton. For all N, defne the proposton: (3.3) P : T Tr B )Y B )X Y )Y (X Y )X. = It s easy to check that, for =, P s true: (3.4) T = Tr B )Y B )X = Tr B )Y B )X Tr (X Y )Y (X Y )X. Now, let us assume that P s true and then prove that ths mples that P s also true. We have that:

4 A TRACE INEQUALITY 3 (3.5) T Tr B )Y B )X = Y )Y (X Y )X. From the recursve formula (3.) we obtan: (3.6) T Tr B )Y B )X = Y )Y (X Y )X Tr B )Y B )X = Tr B )Y B )X Tr B )Y Y )Y (X Y )X Tr B )Y B )X = = Tr B )Y B )X Tr B )Y Y )Y (X Y )X Tr B )Y B )X Y )Y B )X = = Tr B )Y B )X Y )Y (X Y )X Tr B )Y B )X Y )Y B )X (X Y )X (X Y )X Tr B )Y (X Y )X Tr B )Y (X Y )X

5 4 ELENA VERONICA BELMEGA, MARC JUNGERS, AND SAMSON LASAULCE = = Tr B )Y B )X A Y B )Y (X A Y B )X = = Tr B )Y B )X Y )Y (X Y )X The second nequalty follows from applyng Lemma. for the second term on the rght and consderng that X = X A, Y = Y B. The thrd equalty follows from Lemma.. Thus, we have proven the desred result. The second step of the proof s straghtforward. From (3.), t s easy to check that T 0 (all the terms of the form Tr XB XA wth X = X H, A 0, B 0 can be re-wrtten as Tr(NN H ) 0 wth N = A / XB / ). References [] E. V. BELMEGA, S. LASAULCE, and M. DEBBAH, A Trace Inequalty for Postve Defnte Matrces, Journal of nequaltes n pure and appled mathematcs (JIPAM), Vol. 0, No., Artcle 5, 4 pp., (009). [] E. V. BELMEGA, S. LASAULCE, M. DEBBAH, M. JUNGERS, and J. DUMONT, Power allocaton games n wreless networks of mult-antenna termnals, Sprnger Telecommuncatons Systems Journal, VALUETOOLS specal ssue, revsed (mnor revsons), (009). [3] J. ROSEN, Exstence and unqueness of equlbrum ponts for concave n-person games, Econometrca, Vol. 33, pp , (965). [4]. M. ABADIR and J. R. MAGNUS, Matrx Algebra, Cambrdge Unversty Press, New York, USA, pp. 39, pp. 338, (005). Unversté Pars-Sud XI, SUPELEC, Laboratore des Sgnaux et Systèmes, Gf-sur- Yvette, France. E-mal address: belmega@lss.supelec.fr URL: CNRS, ENSEM, CRAN, Vandoeuvre, France. E-mal address: marc.jungers@cran.uhp-nancy.fr URL: CNRS, SUPELEC, Laboratore des Sgnaux et Systèmes, Gf-sur-Yvette, France. E-mal address: lasaulce@lss.supelec.fr URL: