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Trace Inequalities for Matrices Bull. Aust. Math. Soc. 87 (2013), 139–148 doi:10.1017/S0004972712000627 TRACE INEQUALITIES FOR MATRICES KHALID SHEBRAWI ˛ and HUSSIEN ALBADAWI (Received 23 February 2012; accepted 20 June 2012) Abstract Trace inequalities for sums and products of matrices are presented. Relations between the given inequalities and earlier results are discussed. Among other inequalities, it is shown that if A and B are positive semidefinite matrices then tr(AB)k ≤ minfkAkk tr Bk; kBkk tr Akg for any positive integer k. 2010 Mathematics subject classification: primary 15A18; secondary 15A42, 15A45. Keywords and phrases: trace inequalities, eigenvalues, singular values. 1. Introduction Let Mn(C) be the algebra of all n × n matrices over the complex number field. The singular values of A 2 Mn(C), denoted by s1(A); s2(A);:::; sn(A), are the eigenvalues ∗ 1=2 of the matrix jAj = (A A) arranged in such a way that s1(A) ≥ s2(A) ≥ · · · ≥ sn(A). 2 ∗ ∗ Note that si (A) = λi(A A) = λi(AA ), so for a positive semidefinite matrix A, we have si(A) = λi(A)(i = 1; 2;:::; n). The trace functional of A 2 Mn(C), denoted by tr A or tr(A), is defined to be the sum of the entries on the main diagonal of A and it is well known that the trace of a Pn matrix A is equal to the sum of its eigenvalues, that is, tr A = j=1 λ j(A). Two principal properties of the trace are that it is a linear functional and, for A; B 2 Mn(C), we have tr(AB) = tr(BA). Trace inequalities are used in many applications such as control theory, quantum information theory, computational statistics and communication systems; see, for example, [5]. For the theory of trace functionals and their applications the reader is referred to [10]. It is well known that the trace functional is submultiplicative, that is, for positive semidefinite matrices A and B in Mn(C), 0 ≤ tr(AB) ≤ tr A tr B: (1.1) c 2012 Australian Mathematical Publishing Association Inc. 0004-9727/2012 $16.00 139 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 24 Sep 2021 at 14:30:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972712000627 140 K. Shebrawi and H. Albadawi [2] Yang et al. [13] show further that for any integer m, tr(AB)m ≤ (tr A2m)1=2(tr B2m)1=2: (1.2) Clearly, (1.2) implies (1.1) since (tr A2)1=2 ≤ tr A for any matrix A. Also, Ando [1] proved a very strong form of Young’s inequality—it was shown that if A and B are in Mn(C), then there is a unitary matrix U such that 1 1 jAB∗j ≤ U jAjp + jBjq U∗; p q where p; q are positive real numbers satisfying 1=p + 1=q = 1, which immediately gives tr jAjp tr jBjq tr jAB∗j ≤ + : p q Recently, trace inequalities for the product of positive semidefinite matrices A1; A2;:::; Am in Mn(C) were given in [6]: Ym pi 1=pi tr jA1A2 ··· Amj ≤ (tr Ai ) (1.3) i=1 and Xm 1 tr jA A ··· A j ≤ tr Api ; (1.4) 1 2 m p i i=1 i where p1;:::; pm are positive semidefinite real numbers such that 1=p1 + ··· + 1=pm = 1. A special case of (1.3) which generalises (1.2) is: if A and B are positive semidefinite matrices in Mn(C) and p; q are positive real numbers such that 1=p + 1=q = 1, then tr(AB) ≤ (tr Ap)1=p(tr Bq)1=q: (1.5) The main purpose of this paper is to establish trace inequalities for matrices. In Section2 we invoke the majorisation relations for singular values and Hölder’s inequality for positive real numbers to get a general trace inequality which yields some earlier results. In Section3 we give trace inequalities for sums and powers of matrices. 2. Trace inequalities for products of matrices In this section, new forms of Hölder and Young trace inequalities for matrices that generalise (1.3), (1.4) and (1.5) are given. The following result will be helpful in refining earlier results. Theorem 2.1. Let Ai 2 Mn(C) and let pi be a positive real number (i = 1; 2;:::; m) Pm such that i=1 1=pi = 1. Then Ym r Ym tr A ≤ (tr jA jrpi )1=pi for r ≥ 1: (2.1) i i i=1 i=1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 24 Sep 2021 at 14:30:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972712000627 [3] Trace inequalities 141 Proof. It is well known that the trace of a matrix equals the sum of its eigenvalues, so Ym r Xn Ym r Xn Ym tr A = λ A = λr A : i j i j i i=1 j=1 i=1 j=1 i=1 Since the singular values for any matrix are the eigenvalues of its absolute value, Xn Ym Xn Ym λr A = sr A : j i j i j=1 i=1 j=1 i=1 Using the weak majorisation of the product of singular values (see, for example, [7, p. 247]), n m n m X r Y X Y r s j Ai ≤ s j(Ai) : j=1 i=1 j=1 i=1 Now Hölder’s inequality for positive real numbers implies that n m m n m n XY YX 1=pi YX 1=pi r rpi rpi s j(Ai) ≤ s j (Ai) = λ j (jAij) j=1 i=1 i=1 j=1 i=1 j=1 m n m YX 1=pi Y rpi rpi 1=pi = λ j(jAij ) = (tr jAij ) : i=1 j=1 i=1 This completes the proof. A special case of (2.1) is the following. If A, B 2 Mn(C) and p, q are positive real numbers such that 1=p + 1=q = 1, then tr jABjr ≤ (tr jAjrp)1=p(tr jBjrq)1=q; which generalises (1.5). Inequality (2.1) can be considered as a Hölder trace inequality, while the Yong trace inequality can be stated as follows. Corollary 2.2. Let Ai 2 Mn(C) and let pi be a positive real number (i = 1; 2;:::; m) Pm such that i=1 1=pi = 1. Then m m Y r X rpi jAij tr Ai ≤ tr for r ≥ 1: (2.2) p i=1 i=1 i rp1 Proof. Using (2.1) and Young’s inequality for the positive real numbers tr jA1j , rp2 rpm tr jA2j ;:::; tr jAmj , m r m m m rp Y Y X 1 X jAij i rpi 1=pi rpi tr Ai ≤ (tr jAij ) ≤ tr jAij = tr ; p p i=1 i=1 i=1 i i=1 i as required. Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 24 Sep 2021 at 14:30:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972712000627 142 K. Shebrawi and H. Albadawi [4] Corollary 2.3. Let Ai 2 Mn(C) be positive semidefinite and αi be a positive real Pm number (i = 1; 2;:::; m) such that i=1 αi = 1. Then, for r ≥ 1, Ym r Ym tr Aαi ≤ (tr Ar)αi (2.3) i i i=1 i=1 and Ym r Xm tr Aαi ≤ α tr Ar: (2.4) i i i i=1 i=1 αi Proof. Replacing Ai by Ai in (2.1) and letting αi = 1=pi (i = 1; 2;:::; m), we obtain (2.3). Inequality (2.4) follows from (2.3) and Young’s inequality. For r = 1 in (2.3), Ym Ym tr Aαi ≤ (tr A )αi ; i i i=1 i=1 which improves a result of Chen and Wong [3], namely Ym Ym tr Aαi ≤ (tr A )αi : i i i=1 i=1 Remark 2.4. Yang [11] presents the following conjecture. Let Ai be a positive definite operator (1 ≤ i ≤ m) in C1. Does the inequality m m Y m jtr(A1A2 ··· Am)j ≤ tr Ai i=1 hold? For matrices the answer is yes. Indeed, for r = 1 in (2.1), Ym Ym tr A ≤ (tr jA jpi )1=pi : i i i=1 i=1 Letting pi = m (i = 1; 2;:::; m), m 1=m Y m tr jA1A2 ··· Amj ≤ tr Ai : i=1 But jtr(A1A2 ··· Am)j ≤ tr jA1A2 ··· Amj; so m m m Y m jtr(A1A2 ··· Am)j ≤ (tr jA1A2 ··· Amj) ≤ tr Ai : i=1 A special case of (2.2) is the following. For positive semidefinite matrices A and B, jAjrp jBjrq tr jAB∗jr ≤ tr + (2.5) p q Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 24 Sep 2021 at 14:30:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972712000627 [5] Trace inequalities 143 for positive real numbers r, p and q such that r ≥ 1 and 1=p + 1=q = 1. The following theorem introduces an inequality related to (2.5), but the relation between these two forms will be discussed in the next section.
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