On some trace inequalities for positive definite Hermitian matrices

R E S E A R C H

Open Access

On some trace inequalities for positive

definite Hermitian matrices

Houqing Zhou

*_{Correspondence:}

zhouhq2004@163.com Department of Mathematics, Shaoyang University, Shaoyang City, Hunan 422000, China

Abstract

LetAbe a positive definite Hermitian matrix, we investigate the trace inequalities ofA. By using the equivalence of the deformed matrix, according to some properties of positive definite Hermitian matrices and some elementary inequalities, we extend some previous works on the trace inequalities for positive definite Hermitian matrices, and we obtain some valuable theory.

MSC: Primary 15A45; 15A57

Keywords: Hermitian matrix; positive definite; trace inequality

1 Introduction

In 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 theith row andjth column are equal to the complex conjugates of the elements in thejth row andith column. In other words, the matrixAis Hermitian if and only ifA=AT, where

ATdenotes the conjugate transpose of matrixA.

Hermitian matrices play an important role in statistical mechanics [], engineering; in cases such as communication, to describen-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 the identical deformation of matrix, and combined with some elementary inequalities, our purpose 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 Hermitian matrices. In Section , we will quote some lemmas; in Section , which is the main part of the paper, using the properties of Hermitian matrices, we investigate the trace inequalities for positive definite Hermitian matrices.

2 Preliminaries

ByMn,m(F) we denote then-by-mmatrices over a fieldF, usually the real numbersRor the

complex 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 matrices

we replaceMn,nbyMn.

LetA= (aij)∈Mn; we may denote the eigenvalues of Abyλ,λ, . . . ,λn, without loss

of generality, where we letλ≥λ≥ · · · ≥λn. Letσ(A) denote the singular value, and

σ≥σ≥ · · · ≥σn. It is well known that ifAis Hermitian, then all eigenvalues ofAare

real numbers and ifAis unitary, then every eigenvalue ofAhas modulus . The sum of two Hermitian matrices of the same size is Hermitian. IfAis Hermitian, thenAk_{is Hermitian}

for allk= , , . . . . IfAis invertible as well, thenA–_{is Hermitian.}

A Hermitian matrix A∈Mnis said to be positive semidefinite, denoted byA≥, if

(Ax,x)≥ for allx∈Cn_{, and it is called positive definite, denoted byA> , if (Ax,x) > }

for all nonzerox∈Cn_(Cn_{denotes complex vector spaces), where (·) denotes the Euclidean}

inner product onCn_.

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.

LetA∈Mn. Then the trace ofAis given bytrA=

n

i=aii. The trace function has the

following properties.

LetA,B∈Mn,α∈C. Then

() tr(A+B) =trA+trB; () trαA=αtrA; () tr(AB) =tr(BA).

3 Some lemmas

The following lemmas play a fundamental role in this paper.

Lemma .[] LetAi∈Mn(i= , , . . . ,m).Then

tr(AA· · ·Am)≤ n

i=

σi(AA· · ·Am)≤ n

i=

σi(A)σi(A)· · ·σi(Am),

whereσ(Ai)≥σ(Ai)≥ · · · ≥σn(Ai).

Lemma .[] Letαi>  (i= , , . . . ,n),andin=αi≥.Let aij>  (j= , , . . . ,m).Then m

j= aα

ja

α

j · · ·a

αn

nj ≤

_m

j= aj

αm

j= aj

α

· · ·

_m

j= anj

αn

Lemma .[] LetA,A, . . . ,Anbe same size positive definite matrices,andα,α, . . . ,αn

are positive real numbers,andn_i=αi= .Then

tr

n

i= Aαi

i ≤ n

i= αitrAi.

4 Main results

Now, we prove the following theorem.

Theorem . LetAi,Bi(i= , , . . . ,m)be same size positive definite matrices,p> ,and



p+



q = .Then

tr

m

i=

(AiBi)≤

_m

i=

trAp_i



pm

i=

trBq_i



q

.

Proof Since the eigenvalues and traces of positive definite matrices are all positive real numbers, the eigenvalues are equal to the singular values. Then, according to Lemma . and the spectral mapping theorem, we have

m i= trA  p i B  q i ≤ m i= n j= σj A  p i B  q i ≤ m i= n j= σj A  p

i σj

B  q i = m i= n j= λj A  p

i λj

B  q i = m i= n j= λ  p

j (Ai)λ



q

j (Bi). (.)

By using Lemma ., we get

m i= trA  p i B  q i ≤ m i= _n j= λj(Ai)



pn

j= λj(Bi)

 q ≤ m i=

(trAi)



p_(trB

i)  q ≤ _m i=

trAi



pm

i=

trBi



q

LetAi=Api,Bi=Bqi. Then we obtain m

i=

tr(AiBi)≤

_m

i=

trAp_i



pm

i=

trBq_i



q

.

Bytr(A+B) =trA+trB, we have

tr

m

i=

(AiBi)≤

_m

i=

trAp_i



pm

i=

trBq_i



q

.

Thus, the proof is completed.

Next, we give a trace inequality for positive definite matrices.

Theorem . Letαi>  (i= , , . . . ,n)and

n

i=αi= .Ai,Bi,Ci(i= , , . . . ,n)are same

size positive definite matrices.Then

tr

_n

i= Aαi

i + n

i= Bαi

i + n

i= Cαi

i ≤ n i=

tr(Ai+Bi+Ci)

αi

.

Proof Since the trace of a matrix is a linear operation, by using Lemma ., it follows that

tr(n_i=Aαi

i +

n i=B

αi

i +

n i=C

αi

i )

n

i=[tr(Ai+Bi+Ci)]αi

=tr

n

i=

Aαi

i

tr(Ai+Bi+Ci)αi

+ B

αi

i

tr(Ai+Bi+Ci)αi

+ C

αi

i

tr(Ai+Bi+Ci)αi

=tr n i= Ai

tr(Ai+Bi+Ci)

αi +tr n i= Bi

tr(Ai+Bi+Ci)

αi +tr n i= C_i

tr(Ai+Bi+Ci)

αi

≤

n

i=

αitrAi

tr(Ai+Bi+Ci)

+

n

i=

αitrBi

tr(Ai+Bi+Ci)

+

n

i=

αitrCi

tr(Ai+Bi+Ci)

=

n

i=

αi(trAi+trBi+trCi)

tr(Ai+Bi+Ci)

=

n

i=

αi= . (.)

Then we have

tr

_n

i= Aαi

i + n

i= Bαi

i + n

i= Cαi

i ≤ n i=

tr(Ai+Bi+Ci)

αi

.

Theorem . LetAi(i= , , . . . ,n)be same size positive definite matrices.Then we have

the inequality

tr(n_i=Ai)

tr(n_i=Ai)

≤

n

i=

trA_i

trAi

.

Proof Whenn= , 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

. (.)

Suppose that the inequality holds whenn=k,i.e.,

tr(k_i=Ai)

tr(k_i=Ai)

≤

k

i=

trA

i

trAi

.

Then forn=k+ , we have

tr(k_i=+Ai)

tr(k_i=+A_i) =

tr(k_i=Ai+Ak+)

tr(k_i=A_i+A_k+)

≤ tr(

k i=Ai)

tr(k_i=Ai)

+trA



k+

trAk+

≤

k

i=

trA

i

trAi

+trA



k+

trAk+

. (.)

That is, the inequality holds whenn=k+ . Thus we have finished the proof.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

I would like to thank the referees for their valuable comments and important suggestions. Project was supported by Hunan Provincial Natural Science Foundation of China #13JJ3118.

Received: 14 August 2013 Accepted: 23 January 2014 Published:12 Feb 2014

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