B Wang

D Petz

DS Mitrinovic

E Carlen

H Minc

Houqing Zhou

JR Magnus

N Bebiano

NN Chan

R Bellman

RI Oliveira

Z Ulukök

Journal of Inequalities and Applications

Crossref

Zhou Journal of Inequalities and Applications 2014, 2014:64http://www.journalofinequalitiesandapplications.com/content/2014/1/64RESEARCH Open AccessOn some trace inequalities for positivedeﬁnite Hermitian matricesHouqing Zhou**Correspondence:zhouhq2004@163.comDepartment of Mathematics,Shaoyang University, Shaoyang City,Hunan 422000, ChinaAbstractLet A be a positive deﬁnite Hermitian matrix, we investigate the trace inequalitiesof A. By using the equivalence of the deformed matrix, according to some propertiesof positive deﬁnite Hermitian matrices and some elementary inequalities, we extendsome previous works on the trace inequalities for positive deﬁnite Hermitian matrices,and we obtain some valuable theory.MSC: Primary 15A45; 15A57Keywords: Hermitian matrix; positive deﬁnite; trace inequality1 IntroductionIn mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with com-plex entries that is equal to its own conjugate transpose. That is, the elements in the ithrow and jth column are equal to the complex conjugates of the elements in the jth rowand ith column. In other words, the matrix A is Hermitian if and only if A = AT , whereAT denotes the conjugate transpose of matrix A.Hermitian matrices play an important role in statistical mechanics [], engineering; incases such as communication, to describe n-dimensional signal cross-correlation proper-ties, like conjugate symmetry, we can use Hermitian matrices.The earliest study of matrix inequality work in the literature was []. In , Bellman[] proved some trace inequalities for positive deﬁnite Hermitian matrices:() tr(AB) ≤ tr(AB);()  tr(AB)≤ trA + trB;() tr(AB)≤ (trA)  (trB)  .Since then, the problems of the trace inequality for positive deﬁnite (semideﬁnite) Her-mitian matrices have caught the attention of scholars, getting a lot of interesting results.There exists a vast literature that studies the trace (see [–]). In the paper, using theidentical deformation of matrix, and combined with some elementary inequalities, ourpurpose is to derive some new results on the trace inequality for positive deﬁnite Hermi-tian matrices.The rest of this paper is organized as follows. In Section , we will give the relevant deﬁ-nitions and properties of Hermitianmatrices. In Section , we will quote some lemmas; inSection , which is the main part of the paper, using the properties of Hermitian matrices,we investigate the trace inequalities for positive deﬁnite Hermitian matrices.©2014 Zhou; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.Zhou Journal of Inequalities and Applications 2014, 2014:64 Page 2 of 6http://www.journalofinequalitiesandapplications.com/content/2014/1/642 PreliminariesByMn,m(F) we denote the n-by-mmatrices over a ﬁeld F, usually the real numbersR or thecomplex numbersC.Most often, the facts discussed are valid in the setting of the complex-entried matrices, in which caseMn,m(C) is abbreviated asMn,m. In case of square matriceswe replaceMn,n byMn.Let A = (aij) ∈ Mn; we may denote the eigenvalues of A by λ,λ, . . . ,λn, without lossof generality, where we let λ ≥ λ ≥ · · · ≥ λn. Let σ (A) denote the singular value, andσ ≥ σ ≥ · · · ≥ σn. It is well known that if A is Hermitian, then all eigenvalues of A arereal numbers and ifA is unitary, then every eigenvalue ofA hasmodulus . The sumof twoHermitian matrices of the same size is Hermitian. If A is Hermitian, then Ak is Hermitianfor all k = , , . . . . If A is invertible as well, then A– is Hermitian.A Hermitian matrix A ∈ Mn is said to be positive semideﬁnite, denoted by A ≥ , if(Ax,x) ≥  for all x ∈ Cn, and it is called positive deﬁnite, denoted by A > , if (Ax,x) > for all nonzero x ∈ Cn (Cn denotes complex vector spaces), where (·) denotes the Euclideaninner product on Cn.Remark .() The sum of any two positive deﬁnite matrices of the same size is positive deﬁnite.() Each eigenvalue of a positive deﬁnite matrix is a non-negative (positive) real number.() The trace and the determinant of a positive deﬁnite matrix are non-negative(positive) real numbers.() Any principal submatrix of a positive deﬁnite matrix is positive deﬁnite.A Hermitian matrix is positive deﬁnite if and only if all of its eigenvalues are non-negative (positive) real numbers.We will use this fact several times.Let A ∈ Mn. Then the trace of A is given by trA =∑ni= aii. The trace function has thefollowing properties.Let A,B ∈Mn, α ∈C. Then() tr(A +B) = trA + trB;() trαA = α trA;() tr(AB) = tr(BA).3 Some lemmasThe following lemmas play a fundamental role in this paper.Lemma . [] Let Ai ∈Mn (i = , , . . . ,m). Then∣∣tr(AA · · ·Am)∣∣≤ n∑i=σi(AA · · ·Am)≤n∑i=σi(A)σi(A) · · ·σi(Am),where σ(Ai)≥ σ(Ai)≥ · · · ≥ σn(Ai).Lemma . [] Let αi >  (i = , , . . . ,n), and∑ni= αi ≥ . Let aij >  (j = , , . . . ,m). Thenm∑j=aαj aαj · · ·aαnnj ≤( m∑j=aj)α( m∑j=aj)α· · ·( m∑j=anj)αn.Zhou Journal of Inequalities and Applications 2014, 2014:64 Page 3 of 6http://www.journalofinequalitiesandapplications.com/content/2014/1/64Lemma. [] LetA,A, . . . ,An be same size positive deﬁnitematrices, and α,α, . . . ,αnare positive real numbers, and∑ni= αi = . Thentrn∏i=Aαii ≤n∑i=αi trAi.4 Main resultsNow, we prove the following theorem.Theorem . Let Ai, Bi (i = , , . . . ,m) be same size positive deﬁnite matrices, p > , andp +q = . Thentrm∑i=(AiBi)≤( m∑i=trApi) p( m∑i=trBqi) q.Proof Since the eigenvalues and traces of positive deﬁnite matrices are all positive realnumbers, the eigenvalues are equal to the singular values. Then, according to Lemma .and the spectral mapping theorem, we havem∑i=trApi Bqi≤m∑i=n∑j=σj(A pi B qi )≤m∑i=n∑j=σj(A pi )σj(B qi )=m∑i=n∑j=λj(A pi )λj(B qi )=m∑i=n∑j=λpj (Ai)λqj (Bi). (.)By using Lemma ., we getm∑i=trApi Bqi≤m∑i=( n∑j=λj(Ai)) p( n∑j=λj(Bi)) q≤m∑i=(trAi)p (trBi)q≤( m∑i=trAi) p( m∑i=trBi) q. (.)Zhou Journal of Inequalities and Applications 2014, 2014:64 Page 4 of 6http://www.journalofinequalitiesandapplications.com/content/2014/1/64Let Ai =Api , Bi = Bqi . Then we obtainm∑i=tr(AiBi)≤( m∑i=trApi) p( m∑i=trBqi) q.By tr(A +B) = trA + trB, we havetrm∑i=(AiBi)≤( m∑i=trApi) p( m∑i=trBqi) q.Thus, the proof is completed. Next, we give a trace inequality for positive deﬁnite matrices.Theorem . Let αi >  (i = , , . . . ,n) and∑ni= αi = . Ai, Bi, Ci (i = , , . . . ,n) are samesize positive deﬁnite matrices. Thentr[ n∏i=Aαii +n∏i=Bαii +n∏i=Cαii]≤n∏i=[tr(Ai +Bi +Ci)]αi .Proof Since the trace of a matrix is a linear operation, by using Lemma ., it followsthattr(∏ni=Aαii +∏ni=Bαii +∏ni=Cαii )∏ni=[tr(Ai +Bi +Ci)]αi= trn∏i=[ Aαiitr(Ai +Bi +Ci)αi+ Bαiitr(Ai +Bi +Ci)αi+ Cαiitr(Ai +Bi +Ci)αi]= trn∏i=[ Aitr(Ai +Bi +Ci)]αi+ trn∏i=[ Bitr(Ai +Bi +Ci)]αi+ trn∏i=[ Citr(Ai + Bi +Ci)]αi≤n∑i=αi trAitr(Ai +Bi +Ci)+n∑i=αi trBitr(Ai +Bi +Ci)+n∑i=αi trCitr(Ai +Bi +Ci)=n∑i=αi(trAi + trBi + trCi)tr(Ai +Bi +Ci)=n∑i=αi = . (.)Then we havetr[ n∏i=Aαii +n∏i=Bαii +n∏i=Cαii]≤n∏i=[tr(Ai +Bi +Ci)]αi . Now we use mathematical induction to deduce our third result.Zhou Journal of Inequalities and Applications 2014, 2014:64 Page 5 of 6http://www.journalofinequalitiesandapplications.com/content/2014/1/64Theorem . Let Ai (i = , , . . . ,n) be same size positive deﬁnite matrices. Then we havethe inequalitytr(∑ni=Ai)tr(∑ni=Ai)≤n∑i=trAitrAi.Proof When n = , according to () on the ﬁrst page, we have trA · trA · tr(AA)≤  trA · trA ·(trA) (trA) = [trA(trA) ][trA(trA) ]≤ [trA(trA)  ] + [trA(trA)  ]= (trA) trA + (trA) trA . (.)That is, tr(AA)≤ trA · trAtrA+ trA · trAtrA. (.)On the other hand,tr(A +A) = trA +  tr(AA) + trA. (.)Note that(trAtrA+ trAtrA)tr(A +A)=(trAtrA+ trAtrA)(trA + trA)= trA + trA +trA · trAtrA+ trA · trAtrA. (.)So, according to (.), (.), we deduce(trAtrA+ trAtrA)tr(A +A)≥ trA +  tr(AA) + trA = tr(A +A). (.)Hence,tr(A +A)tr(A +A)≤ trAtrA+ trAtrA. (.)Then the inequality holds.Zhou Journal of Inequalities and Applications 2014, 2014:64 Page 6 of 6http://www.journalofinequalitiesandapplications.com/content/2014/1/64Suppose that the inequality holds when n = k, i.e.,tr(∑ki=Ai)tr(∑ki=Ai)≤k∑i=trAitrAi.Then for n = k + , we havetr(∑k+i= Ai)tr(∑k+i= Ai)= tr(∑ki=Ai +Ak+)tr(∑ki=Ai +Ak+)≤ tr(∑ki=Ai)tr(∑ki=Ai)+ trAk+trAk+≤k∑i=trAitrAi+ trAk+trAk+. (.)That is, the inequality holds when n = k + . Thus we have ﬁnished the proof. Competing interestsThe author declares that they have no competing interests.AcknowledgementsI would like to thank the referees for their valuable comments and important suggestions. Project was supported byHunan Provincial Natural Science Foundation of China #13JJ3118.Received: 14 August 2013 Accepted: 23 January 2014 Published: 12 Feb 2014References1. Bebiano, N, Da Providencia, J, Lemos, R: Matrix inequalities in statistical mechanics. Linear Algebra Appl. 376, 265-273(2004)2. Minc, H, Marcus, M: A Survey of Matrix Theory and Matrix Inequalities. Prindle, Weber & Schmidt, Boston (1964)3. Bellman, R: Some inequalities for positive deﬁnite matrices. In: Proceedings of the International Conference onGeneral Inequalities, pp. 89-90. Birkhäuser, Basel (1980)4. Chan, NN, Kwong, MK: Hermitian matrix inequalities and a conjecture. Am. Math. Mon. 92(8), 533-541 (1985)5. Petz, D: A survey of certain trace inequalities. In: Functional Analysis and Operator Theory (Warsaw, 1992). BanachCenter Publications, vol. 30, pp. 287-298. Polish Acad. Sci., Warsaw (1994)6. Carlen, E: Trace inequalities and quantum entropy: an introductory course. In: Entropy and the Quantum, ArizonaSchool of Analysis with Applications, University of Arizona, 16-20 March 2009. Contemporary Mathematics. Am.Math. Soc., Providence (2010)7. Oliveira, RI: Sums of random Hermitian matrices and an inequality by Rudelson. Electron. Commun. Probab. 15,203-212 (2010)8. Ulukök, Z, Türkmen, R: On some matrix trace inequalities. J. Inequal. Appl. 2010, Article ID 201486 (2010)9. Wang, B: Majorization. Beijing Normal University Publishing Group, Beijing (1990)10. Mitrinovic, DS, Vasic, PP: Analytic Inequalities. Springer, Berlin (1970)11. Magnus, JR: A representation theorem for (trAp)1p . Linear Algebra Appl. 95, 127-134 (1987)10.1186/1029-242X-2014-64Cite this article as: Zhou: On some trace inequalities for positive deﬁnite Hermitian matrices. Journal of Inequalitiesand Applications 2014, 2014:64

On some trace inequalities for positive definite Hermitian matrices

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On some trace inequalities for positive definite Hermitian matrices

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