Extension and generalization inequalities involving the Khatri Rao product of several positive matrices

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INVOLVING THE KHATRI-RAO PRODUCT OF

SEVERAL POSITIVE MATRICES

ZEYAD ABDEL AZIZ AL ZHOUR AND ADEM KILICMAN

Received 15 February 2005; Accepted 16 October 2005

Recently, there have been many authors, who established a number of inequalities in-volving Khatri-Rao and Hadamard products of two positive matrices. In this paper, the results are established in the following three ways. First, we find generalization of the inequalities involving Khatri-Rao product using results given by Liu (1999), Mond and Peˇcari´c (1997), Cao et al. (2002), Chollet (1997), and Visick (2000). Second, we recover and develop some results of Visick. Third, the results are extended to the case of Khatri-Rao product of any finite number of matrices. These results lead to inequalities involving Hadamard product, as a special case.

Copyright © 2006 Z. A. Al Zhour and A. Kilicman. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Consider matricesAandBof orderm×nand p×q, respectively. LetA=[Ai j] be par-titioned with Ai j of order mi×nj as the (i,j)th block submatrix and letB=[Bkl] be partitioned withBkl of order pk×ql as the (k,l)th block submatrix (m=ti=1mi, n= d

j=1nj, p= u

k=1pk, q= v

l=1ql). For simplicity, we say that AandB arecompatible partitioned ifA=[Ai j]ti,j=1 andB=[Bi j]ti,j=1 are square matrices of orderm×mand

partitioned, respectively, withAi jandBi jof ordermi×mj(m=ti=1mi=tj=1mj). Let A⊗B, A◦B,AΘB, and A∗B be the Kronecker, Hadamard, Tracy-Singh, and Khatri-Rao products, respectively, ofA and B. The definitions of the mentioned four matrix products are given by Liu in [5,6] as follows:

(i)Kronecker product

A⊗B=ai jB

i j, (1.1)

whereA=[ai j],B=[bkl] are scalar matrices of orderm×nand p×q, respec-tively,ai jBis of orderp×q, andA⊗Bof ordermp×nq;

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 80878, Pages1–21

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(ii)Hadamard product

A◦B=ai jbi j

i j=B◦A, (1.2)

whereA=[ai j],B=[bi j] are scalar matrices of orderm×n,ai jbi jis a scalar, and

A◦Bis of orderm×n; (iii)Tracy-Singh product

AΘB=Ai jΘB

i j=

Ai j⊗Bkl

kl

i j, (1.3)

where A=[Ai j], B=[Bkl] are partitioned matrices of order m×n and p×

q, respectively,Ai j is of ordermi×nj,Bkl of order pk×ql,Ai j⊗Bkl of order

mipk×njql,Ai jΘBof ordermip×njq(m= _t

i=1mi,n= _d

j=1nj, p= _u

k=1pk,

q=v

l=1ql), andAΘBof ordermp×nq; (iv)Khatri-Rao product

A∗B=Ai j⊗Bi j

i j, (1.4)

whereA=[Ai j],B=[Bi j] are partitioned matrices of orderm×nand p×q, respectively,Ai jis of ordermi×nj,Bklof orderpi×qj,Ai j⊗Bi jof ordermipi×

njqj(m= _t

i=1mi,n= _d

j=1nj, p= _t

i=1pi, q= _d

j=1qj), andA∗Bof order

M×N(M=t

i=1mipi,N=dj=1njqj).

In general,AΘB=BΘA,A⊗B=B⊗A,A∗B=B∗A, but ifA=[ai j] is a scalar matrix andB=[Bi j] is a partitioned matrix, thenA∗B=B∗A. Additionally, Liu [5] shows that the Khatri-Rao product can be viewed as a generalized Hadamard product and the Tracy-Singh product as a generalized Kronecker product, as follows:

(1) for a nonpartitioned matrixA, theirAΘBisA⊗B, that is,

AΘB=ai jΘB

i j=

ai j⊗Bkl

kl

i j=

ai jBkl

kl

i j=

ai jB

i j=A⊗B; (1.5)

(2) for nonpartitioned matricesAandBof orderm×n, theirA∗BisA◦B, that is,

A∗B=ai j⊗bi j

i j=

ai jbi j

i j=A◦B. (1.6)

The Khatri-Rao and Tracy-Singh products are related by the following relation [5,6]:

A∗B=ZT

1(AΘB)Z2, (1.7)

whereA=[Ai j] is partitioned withAi jof ordermi×njandB=[Bkl] is partitioned with

Bklof orderpk×ql(m=it=1mi,n=dj=1nj, p=uk=1pk,q=vl=1ql),Z1is anmp× r(r=t

i=1mipi) matrix of zeros and ones, andZ2 is annq×s(s=

d

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of zeros and ones such thatZT1Z1=Ir,Z2TZ2=Is (Ir andIsarer×r ands×s identity matrices, resp.).

In particular, ifm=nandp=q, then there exists anmp×r(r=t

i=1mipi) matrixZ such thatZT_Z₌_I

r(Iris anr×ridentity matrix) and

A∗B=ZT(AΘB)Z. (1.8)

Here

Z= ⎡ ⎢ ⎢ ⎣

Z1

. ..

Zt ⎤ ⎥ ⎥

⎦, (1.9)

where eachZi=[0i1···0ii−1I_{mi pi} 0i i+1···0it]T is an real matrix of zeros and ones, and 0_ik is

amipi×mipkzero matrix for anyk=i. Note also thatZiTZi=Iand

Z_iTAi jΘBZj=ZiT

Ai j⊗BklklZj=Ai j⊗Bi j, i,j=1, 2,. . .,t. (1.10)

In [5–8], the authors proved a number of equalities and inequalities involving Khatri-Rao and Hadamard products of two matrices. Here we extend these results in three ways. First, we establish new attractive equalities and inequalities involving Khatri-Rao prod-uct of matrices. Second, we recover and develop some results of Visick, for example, [8, Theorem 11, page 54]. This does not follow simply from the work of Visick. Third, the results are extended to the case of Khatri-Rao products of any finite number of matrices. This result leads to inequalities involving Hadamard product, as a special case.

We use the following notations:

(i)Mm,n—the set of allm×nmatrices over the complex number fieldCand when

m=n, we writeMminstead ofMm,n;

(ii)AT_,_A∗_,_A+_,_A−1_{—the transpose, conjugate transpose, Moore-Penrose inverse,}

and inverse of matrixA, respectively.

For Hermitian matricesAandB, the relationA > Bmeans thatA−B >0 is a positive definite and the relationA≥BmeansA−B≥0 is a positive semidefinite. Given a positive definite matrixA, its positive definite square root is denoted byA1/2_{. We use the known}

fact “for positive definite matricesAandB, the relationA≥BimpliesA1/2_≥_B1/2_{” which}

is called theL¨owner-Heinz theorem.

2. Some notations and preliminary results

LetAbe a positive definitem×mmatrix. Thespectral decompositionof matrixAassures that there exists a unitary matrixUsuch that

A=U∗DU=U∗diagλi

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whereD=diag(λi)=diag(λ1,. . .,λm) is the diagonal matrix with diagonal entriesλi(λi are the positive eigenvalues ofA). For any real numberr,Ar_{is defined by}

Ar=U∗DrU=U∗diagλr_iU. (2.2)

IfA∈Mm,nis any matrix with rank (A)=s, thesingular value decompositionofAassures that there are unitary matricesU∈MmandV∈Mnsuch that

A=UV∗. (2.3)

Here=[W0 00]∈Mm,n, whereW=diag(σ1,. . .,σs)∈Msis the diagonal matrix with di-agonal entriesσi (i=1, 2,. . .,s) andσ1≥σ2≥ ··· ≥σs>0 are the singular values ofA, that is,σ1≥σ2≥ ··· ≥σs>0 are positive square roots of positive eigenvalues ofA∗Aand

AA∗. TheMoore-Penrose inverseofAis defined by

A+₌_V

W−1 ₀

0 0

U∗∈Mn,m, (2.4)

where W−1₌_diag(_σ−1

1 ,σ−21,. . .,σ−s1)∈Ms is the diagonal matrix with diagonal entries

σ−1

i (i=1, 2,. . .,s).A+is a unique matrix which satisfies the following conditions:

AA+_A₌_A_, _A+_AA+₌_A+_, _AA+∗₌_AA+_, _A+_A∗₌_A+_A. _(2.5)

For any compatible partitioned matricesA,B,C, andD, we will make a frequent use of the following properties of the Tracy-Singh product (see e.g., [1,3,5,10]):

(a) (AΘB)(CΘD)=(AC)Θ(BD) ifACandBDare well defined; (b) (AΘB)r₌_Ar_Θ_Br _if_A_∈_M

m, B∈Mnare positive semidefinite matrices andris any real number;

(c) (AΘB)∗=A∗ΘB∗; (d) (AΘB)+₌_A+_Θ_B+_.

IfA∈MmandB∈Mnare positive semidefinite matrices, then (see, [3,10]) (e)AΘB≥0;

(f)λ1(AΘB)=λ1(A)λ1(B), λmn(AΘB)=λm(A)λn(B),

whereλ1(A),λm(A) are the largest and smallest eigenvalues, respectively, of a matrixA, andλ1(B),λn(B) are the largest and smallest eigenvalues, respectively, of a matrixB.

The Khatri-Rao and Tracy-Singh products ofkmatricesAi(1≤i≤k,k≥2) will be denoted byk_i₌1∗Ai=A1∗A2∗ ··· ∗Ak and ki=1ΘAi=A1ΘA2Θ···ΘAk, respec-tively.

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Lemma2.1. LetAiandBi(1≤i≤k,k≥2)be compatible partitioned matrices. Then (i)

_k

i=1

ΘAi

_k

i=1

ΘBi

=

_k

i=1

ΘAiBi

(2.6)

ifAiBi(1≤i≤k,k≥2)are well defined; (ii)

_k

i=1

ΘAi +

= k

i=1

ΘA+

i, k=2, 3,. . .; (2.7)

(iii)

_k

i=1

ΘAi ∗

= k

i=1

ΘA∗_i ,

_k

i=1

∗Ai ∗

= k

i=1

∗A∗_i, k=2, 3,. . .; (2.8)

(iv)

_k

i=1

ΘAi r

= k

i=1

ΘAr

i ifAi∈Mm(i)(1≤i≤k,k≥2) (2.9)

are positive semidefinite matrices andris any real number; (v)

_k

i=1

AiΘBi

=

_k

i=1 Ai

Θ

_k

i=1 Bi

, k=2, 3,. . . . (2.10)

Proof. The proof is immediately derived by induction onk.

Lemma 2.2. Let Ai=[A(ghi)]∈Mm(i),n(i) (1≤i≤k, k≥2)be partitioned matrices with A(_ghi) as the(g,h)th block submatrix(m=k

i=1m(i), n=

k

i=1n(i), r=

t j=1

k i=1mj(i),

s=t j=1

k

i=1nj(i), m(i)=tj=1mj(i), n(i)=tj=1nj(i)). Then there exist two real ma-tricesZ1of orderm×randZ2of ordern×ssuch thatZ1TZ1=Ir,Z2TZ2=Is(Z1,Z2are real

matrices of zeros and ones) and

k

i=1

∗Ai=Z1T

_k

i=1

ΘAi

Z2, k=2, 3,. . ., (2.11)

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m(i)=n(i) (1≤i≤k,k≥2), then there exists anm×rmatrixZ of zeros and ones such thatZT_Z₌_I

r,

k

i=1

∗Ai=ZT

_k

i=1

ΘAi

Z, k=2, 3,. . ., (2.12)

andZZT _{is an}_m_×_m_{diagonal matrix of zeros and ones, so}

0≤ZZT_≤_I

m, (2.13)

wherem=k i=1m(i).

Proof. The special case in (2.12) ofLemma 2.2is proved in [3, Corollary 2.2] and (2.13) follows immediately by the definition of matrixZ. We give proof of the general case in (2.11) ofLemma 2.2for the sake of convenience. We proceed by induction onk. Ifk=2, then (2.11) is true by (1.7). Now suppose (2.11) holds for the Khatri-Rao product ofk

matrices, that is, there exist anm×rmatrixPkrof zeros and ones and ann×smatrixRks of zeros and ones such thatP_krTPkr=Ir,RTksRks=Is, and

k

i=1

∗Ai=PkrT

_k

i=1

ΘAi

Rks, k=2, 3,. . . . (2.14)

We will prove that it is true for the Khatri-Rao product ofk+ 1 matrices. Then by (1.7), there exist anm(1)r×rmatrixQ1of zeros and ones and ann(1)s×smatrixQ2of zeros

and ones such thatQT

1Q1=Ir,QT2Q2=Is, and k+1

i=1

∗Ai=A1∗

_k+1

i=2

∗Ai

=QT

1

A1Θ

k+1

i=2

∗Ai

Q2=QT1

A1Θ

PT kr

_k+1

i=2

ΘAi

Rks

Q2

=QT

1

Im(1)A1In(1)

Θ

PT kr

_k+1

i=2

ΘAi

Rks

Q2

=QT1

Im(1)ΘP_krT

A1Θ

_k+1

i=2

ΘAi

In(1)ΘRks

Q2

=QT1

Im(1)ΘP_krT

k+1

i=1

ΘAi

In(1)ΘRks

Q2.

(2.15)

LettingZ1=(Im(1)ΘPkr)Q1andZ2=(In(1)ΘRks)Q2, the inductive step is complete. Here Q1=P2r=Pr,Q1=R2s=Rs, and it is a simple matter to verify that

Z1=Im(1)ΘPkrPr=P(k+1)r, Z1T=PrT

Im(1)ΘPT_kr=PT(k+1)r,

Z2=

In(1)ΘRks

Rs=R(k+1)s, Z2T=RTs

In(1)ΘRT_ks

=RT

(k+1)s.

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Note that

ZT

1Z1=PTr

Im(1)ΘP_krT

Im(1)ΘPkr

Pr=Q1T

Im(1)ΘP_krT

Im(1)ΘPkr

Q1

=QT1

Im(1)Im(1)ΘPT_krPkr

Q1

=QT

1

Im(1)ΘIr

Q1

Im(1)ΘIr=Im(1)r

=QT

1

Im(1)r

Q1=QT1Q1=Ir.

(2.17)

Similarly, it is easy to verify thatZT

2Z2=Is.

Lemma2.3. Letαbe a nonempty subset of the set{1, 2,. . .,m}and letA∈Mmbe a positive semidefinite matrix. Then (see Chollet [4])

(i)if either−1≤r≤0or1≤r≤2, then

Ar₍_α₎_≥_A₍_α₎r_, _∀_α_; _(2.18)

(ii)if0≤r≤1, then

Ar₍_α₎_≤_A₍_α₎r_, _∀_α_, _(2.19)

whereA(α)is the principal submatrix ofAwhose entries are in the intersection of the rows and columns ofAspecified byα.

Lemma2.4. LetXj>0 (j=1, 2,. . .,k)ben×nmatrices with eigenvalues in the interval [w,W]andUj (j=1, 2,. . .,k)are r×m matrices such thatkj=1UjU∗j =I. Then (see Mond and Peˇcari´c [7])

(i)for every realp >1andp <0,

k

j=1

UjXjpU∗j ≤μ

_k

j=1

UjXjU∗j p

, (2.20)

where

μ= δp−δ

(p−1)(δ−1)

p−1

p

δp₋₁

δp₋_δ p

, δ=W

w. (2.21)

While for0< p <1, the reverse inequality holds in (2.20); (ii)for every realp >1andp <0,

_k

j=1

UjXpjU∗j

−

_k

j=1

UjXjU∗j p

≤γ{I}, (2.22)

where

γ=Wwp−wWp

W−w + (p−1)

1

p

Wp₋_wP

W−w

p/(p−1)

. (2.23)

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3. New applications and results

Based on the basic results inSection 2and the general connection between the Khatri-Rao and Tracy-Singh products inLemma 2.2, we generalize and derive some equalities and inequalities in works of Visick [8, Corollary 3, Theorem 4], Chollet [4], and Mond and Peˇcari´c [7] with respect to the Khatri-Rao product and extend these results to any finite number of matrices. These results lead to inequalities involving Hadamard products, as a special case.

Theorem3.1. LetAi=[A(ghi)]∈Mm(i),n(i)(1≤i≤k,k≥2)be partitioned matrices with A(_ghi)as the(g,h)th block submatrix(m=k

i=1m(i),n=

_k

i=1n(i))and letZ1andZ2be the

real matrices of zeros and ones that satisfy (2.11). Then

(i)there exists anm×(m−r)matrixQ(m)of zeros and ones such that the block matrix

Ω=[Z1Q(m)]is anm×mpermutation matrix.Q₍_m₎is not unique but for any such choice ofQ(m),

Z1TQ(m)=0, QT(m)Q(m)=Im−r, Q(m)QT(m)+Z1Z1T=Im (3.1)

(ii)for anym×nmatrixL,

ZT

1LL∗Z1≥

ZT

1LZ2

ZT

1LZ2

∗_≥₀_. _(3.2)

Proof. Though the proof is quite similar to the proof of [8, Corollary 3(iii) and (vii)] for Hadamard product, we give proof for the sake of convenience.

(i) It is evident from the structure ofZ1that it may be considered as part of anm×m

permutation matrixΩ=[Z1Q(m)], whereQ₍_m₎is anm×(m−r) matrix of zeros and ones. For example, whenk=2, thenQ(2)is not unique (see, [8, page 49]). Using the properties

of a permutation matrix together with the definition ofΩ=[Z1Q(m)], we have

Im=ΩΩT=

Z1 Q(m)

_ZT

1 QT(m)

=Q(m)Q(Tm)+Z1Z1T,

Im=

Ir 0

0 Im−r

=ΩT_Ω₌

ZT

1 QT m

Z1 Q(m)

=

ZT

1Z1 Z1TQ(m) Q(Tm)Z1 QT(m)Q(m)

.

(3.3)

From these come the required results in (i), that is,

ZT

1Q(m)=0, Q(Tm)Q(m)=Im−r, Q(m)QT(m)+Z1Z1T=Im. (3.4)

(ii) By (2.13) ofLemma 2.2, we haveIn≥Z2Z2T≥0 and so

Z1TLL∗Z1≥Z1TLZ2Z2TL∗Z1=Z1TLZ2Z1TLZ2∗≥0. (3.5)

We now generalize [8, Theorem 4] to the case of Khatri-Rao product involving a finite

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Theorem3.2. LetAi=[A(ghi)]∈Mm,n(1≤i≤k,k≥2)be partitioned matrices withA(ghi) as the(g,h)th block submatrix. LetZ1be anmk×rmatrix of zeros and ones that satisfies

(2.12) and letQ(n)be annk×(nk−s)matrix of zeros and ones that satisfies (3.1). Then

k

i=1

∗(AiA∗i)=

_k

i=1

∗(Ai)

_k

i=1

∗Ai ∗

+ZT1

_k

i=1

ΘAi

Q(n)QT(n)

_k

i=1

ΘAi ∗

Z1

=

_k

i=1

∗(Ai)

_k

i=1

∗Ai ∗ + ZT 1 _k

i=1

ΘAi

Q(n)

ZT

1

_k

i=1

ΘAi

Q(n)

∗ , (3.6) and hence k

i=1

∗AiA∗i

≥

_k

i=1

∗(Ai)

_k

i=1

∗Ai ∗

, k=2, 3,. . . . (3.7)

Proof. FromLemma 2.1(i) and (iii), we have

k

i=1

ΘAiA∗i

=

_k

i=1

ΘAi

_k

i=1

ΘAi ∗

. (3.8)

But byTheorem 3.1(i), there exist annk_×_s _matrix_Z

2 of zeros and ones that satisfies

(2.12) and an nk_×₍_nk₋_s_{) matrix}_Q

(n) of zeros and ones that satisfies (3.1) such that Z2Z2T+Q(n)QT(n)=Inkand

k

i=1

ΘAiA∗i

=

_k

i=1

ΘAi

Z2Z2T+Q(n)Q(Tn)

k

i=1

ΘAi ∗

=

_k

i=1

ΘAi

Z2Z2T

k

i=1

ΘAi ∗

+

_k

i=1

ΘAi

Q(n)QT(n)

k

i=1

ΘAi ∗

.

(3.9)

SinceAi(1≤i≤k,k≥2) are rectangular partitioned matrices of orderm×n, then due to (2.11) ofLemma 2.2there exist two real matricesZ1andZ2of zeros and ones of order mk_×_r_and_nk_×_s_{, respectively, such that}

k

i=1

∗Ai=ZT1

_k

i=1

ΘAi

Z2, k=2, 3,. . . . (3.10)

But becauseAiA∗i (1≤i≤k, k≥2) are square matrices of orderm×m, then due to (2.12) ofLemma 2.2there exists a real matrixZ1of zeros and ones of ordermk×rsuch

that

k

i=1

∗AiA∗i

=ZT

1

_k

i=1

ΘAiA∗i

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Due to (3.9), (3.10), and (3.11), we have

k

i=1

∗AiA∗i

=Z1T

_k

i=1

ΘAiA∗i

Z1=Z1T

_k

i=1

ΘAi

Z2Z2T

_k

i=1

ΘAi ∗

Z1

+Z1T

_k

i=1

ΘAi

Q(n)Q(Tn)

k

i=1

ΘAi ∗ Z1 = ZT 1 _k

i=1

ΘAi Z2 ZT 1 _k

i=1

ΘAi

Z2

∗

+ZT

1

_k

i=1

ΘAi

Q(n)Q(Tn)

k

i=1

ΘAi ∗

Z1

=

_k

i=1

∗Ai

k

i=1

∗Ai ∗

+Z1T

_k

i=1

ΘAi

Q(n)QT(n)

_k

i=1

ΘAi ∗

Z1

=

_k

i=1

∗Ai

k

i=1

∗Ai ∗

+

Z1T

_k

i=1

ΘAi

Q(n)

Z1T

_k

i=1

ΘAi

Q(n)

∗

.

(3.12) If we putk=2 inTheorem 3.2, we obtain the following corollary.

Corollary3.3. LetAi=[A(ghi)]∈Mm,n(1≤i≤2)be partitioned matrices withA(ghi)as the (g,h)th block submatrix. LetZ1be anm2×r matrix of zeros and ones that satisfies (1.8)

and letQ(n)be ann2×(n2−s)matrix of zeros and ones that satisfies (3.1). Then

A1A∗1 ∗A2A∗2 =

A1∗A2

∗₊_ZT

1

A1ΘA2

Q(n)QT(n)

A1ΘA2

∗_Z

1, (3.13)

and hence

A1A∗1 ∗A2A∗2 ≥

A1∗A2

∗_.

(3.14)

Corollary3.4. LetAi=[A(ghi)]∈Mm,n(1≤i≤k,k≥2)be partitioned matrices withA(ghi) as the(g,h)th block submatrix. LetZ1be anmk×rmatrix of zeros and ones that satisfies

(2.12) and letQ(n)be annk×(nk−s)matrix of zeros and ones that satisfies (3.1). Then the

following statements are equivalent: (i)

k

i=1

∗AiA∗i

=

_k

i=1

∗Ai

k

i=1

∗Ai ∗

, k=2, 3,. . .; (3.15)

(ii)

ZT

1

_k

i=1

ΘAi

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(iii)

k

i=1

∗AiXi=

_k

i=1

∗Ai

_k

i=1

∗Xi

, forXi∈Mn,m(1≤i≤k,k≥2). (3.17)

Proof. To arrive from (i) to (ii), notice that (i) holds if and only if the last term of (3.6) is zero, which is equivalent toZ1T(

k

i=1ΘAi)Q(n)=0. To arrive from (ii) to (iii), notice

that (ii) may be rewritten asZ1T(

k

i=1ΘAi)Q(n)Q(Tn)=0. ByTheorem 3.1(i), there exist an nk_×_s_matrix_Z

2of zeros and ones that satisfies (2.12) and annk×(nk−s) matrixQ(n)of

zeros and ones that satisfies (3.1) such thatQ(n)QT(n)=Ink−Z2Z2T, this becomes

Z1T

_k

i=1

ΘAi

=Z1T

_k

i=1

ΘAi

Z2ZT2. (3.18)

By postmultiplying by (ki=1ΘXi)Z1for any of then×mmatricesXi(1≤i≤k), we have

Z1T

_k

i=1

ΘAiXi

Z1=Z1T

_k

i=1

ΘAi

Z2Z2T

_k

i=1

ΘXi

Z1, (3.19)

which is (iii) by (2.11) and (2.12) ofLemma 2.2. To arrive from (iii) to (i), assume (iii) holds for alln×mmatricesXi(1≤i≤k). It must therefore be true forXi=A∗i (1≤i≤

k), which is condition (i). Hence (iii) implies (3.6) which is (i). If we putk=2 inCorollary 3.4, we obtain the following corollary.

Corollary3.5. LetAi=[A(ghi)]∈Mm,n(1≤i≤2)be partitioned matrices withA(ghi)as the (g,h)th block submatrix. LetZ1be anm2×rmatrix of zeros and ones that satisfies (1.8) and

letQ(n)be ann2×(n2−s)matrix of zeros and ones that satisfies (3.1). Then the following

statements are equivalent: (i)

A1A∗1 ∗A2A∗2 =

A1∗A2

∗

; (3.20)

(ii)

ZT

1

A1ΘA2

Q(n)=0; (3.21)

(iii)

A1X1∗A2X2=

A1∗A2

X1∗X2

, forX1,X2∈Mn,m. (3.22)

Theorem3.6. LetAi≥0 (1≤i≤k,k≥2)ben×ncompatible partitioned matrices. Then (i)if either−1≤r≤0or1≤r≤2, then

k

i=1

∗Ar i≥

_k

i=1

∗Ai r

(12)

(ii)if0≤r≤1, then

k

i=1

∗Ar i≤

_k

i=1

∗Ai r

. (3.24)

Proof. If we puts=1, replacerby 1/randAibyAri in [3, Theorem 3.1(i)], we obtain (i). But, if we puts= −1, replacerby 1/−randAibyA−irin [3, Theorem 3.1(i)], we obtain

(ii).

Remark 3.7. It is easy to give another proof ofTheorem 3.6by replacingAbyki=1ΘAi inLemma 2.3and applying (2.12) ofLemma 2.2.

Theorem3.8. LetAi>0be compatible partitioned matrices such thatki=1ΘAi>0 (1≤

i≤k,k≥2). LetWandwbe the largest and smallest eigenvalues ofk_i₌1ΘAi, respectively. Then

(i)for every realp >1andp <0,

k

i=1

∗Aip≤μ

_k

i=1

∗Ai p

, k=2, 3,. . ., (3.25)

where

μ= δp−δ

(p−1)(δ−1)

p−1

p

δp₋₁

δp₋_δ p

, δ=W

w. (3.26)

While for every0< p <1, the reverse inequality holds in (3.25); (ii)for every realp >1andp <0,

k

i=1

∗A_ip−

_k

i=1

∗Ai p

≤γI, k=2, 3,. . ., (3.27)

where

γ=Wwp−wWp

W−w + (p−1)

1

p

Wp₋_wP

W−w

p/(p−1)

. (3.28)

While for every0< p <1, the reverse inequality holds in (3.27).

Proof. This theorem follows from [3, Theorem 3.1(ii) and (iii)]. We give proof for the sake of convenience. In (2.20) and (2.22) ofLemma 2.4, setk=1 and replaceU byZT_,

U∗ byZ, and X by_ik₌1ΘAi, whereZ, is the selection matrix of zeros and ones that satisfies (2.12). By usingLemma 2.1(iv), we establishTheorem 3.8.

From (3.25), we have the following special cases: (i) forp=2, we have

k

i=1

∗A2

i ≤

(W+w)2

4wW

_k

i=1

∗Ai 2

(13)

(ii) forp= −1, we have

k

i=1

∗A−1

i ≤

(W+w)2

4wW

_k

i=1

∗Ai −1

, k=2, 3,. . . . (3.30)

From (3.27), we have the following special cases: (i) forp=2, we have

k

i=1

∗A2_i−

_k

i=1

∗Ai 2

≤1

4(W−w)

2_{_I_}_, _k₌_{2, 3,}_{. . .}_; _(3.31)

(ii) forp= −1, we have

k

i=1

∗A−i1−

_k

i=1

∗Ai −1

≤ √

W−√w wW

I, k=2, 3,. . . . (3.32)

4. Further developments and applications

Due toAlbert’s theoremin [2] and [9, Theorem 6.13], for a partitioned matrix [_BA B∗_D] with a positive (semi) definite matrixA∈Mm,

A B

B∗ D

≥0 iﬀD≥B∗A+_B_, _(4.1)

for any positive semidefinite matrixD∈Mn. It is also known that if matrixAis square and nonsingular, thenA+₌_A−1_{and [}A B

B∗ _D]≥0 if and only ifD≥B∗A−1B.

LetZ1andZ2be the real matrices of zeros and ones of orderm×randn×s,

respec-tively, that satisfy (2.11) inLemma 2.2. Now another way to useLemma 2.2to generate inequalities involving the Khatri-Rao product is by using the following obvious inequal-ity:

TT∗=

T1 T2

T1∗ T2∗

=

T1T1∗ T1T2∗ T2T1∗ T2T2∗

≥0, (4.2)

whereT1andT2aren×landm×lmatrices, respectively. Note thatT1T1∗andT2T2∗are

positive semidefinite (positive definite) matrices for every (nonsingular) complex matri-cesT1andT2. This leads to

Z2T 0

0 ZT

1

T1T1∗ T1T2∗ T2T1∗ T2T2∗

Z2 0

0 Z1

=

Z2TT1T1∗Z2 Z2TT1T2∗Z1 ZT

1T2T1∗Z2 Z1TT2T2∗Z1

≥0, (4.3)

if and only if

ZT

1T2T2∗Z1≥

ZT

1T2T1∗Z2

ZT

2T1T1∗Z2

+ ZT

2T1T2∗Z1

(14)

Therefore (4.4) can be considered to be more general than (3.2). In order to prove this we setT1=IandT2=Lin (4.4), we have

ZT

1LL∗Z1≥

ZT

1LI∗Z2

ZT

2II∗Z2

+ ZT

2IL∗Z1

=Z1TLZ2Z2TZ2+Z2TL∗Z1 (Z2TZ2=I)

=ZT

1LZ2

ZT

1LZ2

∗_.

(4.5)

Returning to (4.4) and (3.2), it can be easily seen that various other choices of the matricesT1,T2, andLare possible which lead to quite diﬀerent inequalities involving

Khatri-Rao products. However, there exist some inequalities that do not seem to follow directly from (1.7) or (2.11), but follow easily from (4.4) and (3.2). Based on (4.4) and (3.2) we generalize some inequalities in works of Visick [8, Corollary 13, Remark in page 56, Theorems 11, 17, and 20] and establish some new inequalities involving Khatri-Rao products of several positive matrices.

Theorem4.1. LetA1andA2be compatible partitioned matrices. Then

A1A∗1∗A2A∗2+A2A∗2 ∗A1A∗1+A1A∗2 ∗A2A∗1 +A2A∗1 ∗A1A∗2

≥A1∗A2+A2∗A1

A1∗A2

∗

+A2∗A1

∗

. (4.6)

Proof. SetT1=IΘIandT2=A1ΘA2+A2ΘA1. Then calculations show that

T2T2∗=A1A∗1ΘA2A∗2 +A2A∗2ΘA1A∗1 +A1A∗2ΘA2A∗1+A2A∗1ΘA1A∗2, T2T1∗=A1ΘA2+A2ΘA1, T1T2∗=

A1ΘA2

∗

+A2ΘA1

∗

, T1T1∗=IΘI.

(4.7)

Substituting these into (4.4) and using (1.7), we get (4.6).

Corollary4.2. LetAi(1≤i≤2)be Hermitian compatible partitioned matrices. Then (i)

A21∗A22≥

A1∗A2

2

; (4.8)

(ii)

A2_∗_A−2_≥_A_∗_A−12 _if_A_{is nonsingular;} _(4.9)

(iii)

I∗A2_≥_I_∗_A2_. _(4.10)

Proof. (i) SetA∗1 =A1andA∗2 =A2in (3.14) ofCorollary 3.3, we get (4.8).

(ii) SetA1=AandA2=A−1in (4.8), we get (4.9).

(iii) SetA1=IandA2=Ain (4.8), we get (4.10).

Corollary4.3. LetAi>0 (1≤i≤2)be compatible partitioned matrices. Then

A2 1∗A22

1/2

(15)

Proof. It follows immediately by (4.8) andL¨owner-Heinz theorem.

Theorem 4.4. Let Ai≥0 (1≤i≤k, k≥2) be compatible partitioned matrices and let

A0i =A1i/2A+1i /2=A+1i /2A1i/2(1≤i≤k). Then

2

_k

i=1

∗A0_i

+

A1∗

k

i=2

∗A+_i

+

A+1∗

k

i=2

∗Ai

≥

A1∗

k

i=2

∗A0

i+A01∗

k

i=2

∗Ai

_k

i=1

∗Ai

+

A1∗

k

i=2

∗A0

i+A01∗

k

i=2

∗Ai

.

(4.12)

Proof. Since Ai≥0 (1≤i≤k, k≥2), then A∗i =Ai. Set T1=ki=1ΘA1i/2 and T2= A11/2Θ

k

i=2ΘA+1i /2+A+11 /2Θ

k

i=2A1i/2. Since Ai1/2A1i/2=Ai,A+1i /2Ai+1/2=A+i, and A0i =

Ai1/2A+1i /2=Ai+1/2A1i/2(1≤i≤k), then calculations show that

T2T2∗=2

_k

i=1

ΘA0i

+

A1Θ

k

i=2

ΘA+i

+

A+1Θ

k

i=2

ΘAi

, T1T1∗=

k

i=1

ΘAi,

T2T1∗=

A1Θ

k

i=2

ΘA0_i+A0 1Θ

k

i=2

ΘAi

, T1T2∗=

A1Θ

k

i=2

ΘA0_i+A0 1Θ

k

i=2

ΘAi

.

(4.13)

Substituting these into (4.4) and usingLemma 2.2, we get (4.12). If we putk=2 and replaceAibyAri(1≤i≤2) inTheorem 4.4, we obtain the following theorem.

Theorem4.5. LetA1≥0,A2≥0be compatible partitioned and letrbe any nonzero real

number such thatA01=Ar/12A1+r/2=A+1r/2Ar/12andA02=Ar/22A2+r/2=A+2r/2Ar/22. Then

2A0

1∗A02+Ar1∗A2+r+A+1r∗Ar2

≥Ar

1∗A02+A01∗Ar2

Ar

1∗Ar2

+ Ar

1∗A02+A01∗Ar2

. (4.14)

IfA1>0,A2>0 inTheorem 4.5, we obtain the following theorem.

Theorem4.6. LetA1>0,A2>0be compatible partitioned and letIbe a compatible

parti-tioned identity matrix. Then for any nonzero real numberr,

2I+Ar1∗A2−r+A−1r∗Ar2≥

Ar1∗I+I∗Ar2

Ar1∗Ar2

−1

Ar1∗I+I∗Ar2

. (4.15)

If we putr=1 andA1=A2inTheorem 4.6, we obtain the following theorem. Theorem4.7. LetA >0be compatible partitioned and letI be a compatible partitioned identity matrix. Then

(16)

In particular, ifIis a nonpartitioned identity matrix, then

2I+A∗A−1+A−1∗A≥4(I∗A)(A∗A)−1(I∗A). (4.17)

Theorem 4.8. Let A1>0 andA2>0 be compatible partitioned matrices. Then for any

nonzero real numberr

A1r∗A−2r+A1−r∗Ar2+ 2I≥

Ar/12∗A−2r/2+A−1r/2∗Ar/22

2

. (4.18)

In particular, ifA1=A2=A, Then

Ar_∗_A−r₊_A−r_∗_Ar_{+ 2}_I_≥_Ar/2_∗_A−r/2₊_A−r/2_∗_Ar/22_. _(4.19)

Proof. Since A1 >0 and A2 >0, then A∗1 =A1 and A∗2 =A2. Set L=A1r/2ΘA−2r/2+ A−1r/2ΘAr/22. Compute

ZT

1LL∗Z1=Z1TLLZ1=Z1T

Ar/2

1 ΘA−2r/2+A1−r/2ΘAr/22

Ar/2

1 ΘA−2r/2+A1−r/2ΘAr/22

Z1

=Z1T

Ar1ΘA−2r

Z1+Z1T(IΘI)Z1+Z1T(IΘI)Z1+Z1T

A−1rΘAr2

Z1

=Ar1∗A−2r+ 2I+A−1r∗Ar2.

(4.20)

Similarly,

ZT

1LZ2

ZT

1LZ2

∗ =ZT

1LZ2

2

=ZT

1

Ar/2

1 ΘA−2r/2+A1−r/2ΘAr/22

Z2

2

=Ar/12∗A2−r/2+A−1r/2∗Ar/22

2

. (4.21)

Substituting (4.20) and (4.21) into (3.2), we get (4.18).

From (4.18), we have the following special cases: (i) forr=1, we have

A1∗A−21+A1−1∗A2+ 2I≥

A11/2∗A−21/2+A−11/2∗A12/2

2

; (4.22)

(ii) forr=2, we have

A2

1∗A−22+A1−2∗A22+ 2I≥

A1∗A−21+A−11∗A2

2

(17)

From (4.19), we have the following special cases: (i) forr=1, we have

A∗A−1₊_A−1_∗_A_{+ 2}_I_≥_A1/2_∗_A−1/2₊_A−1/2_∗_A1/22_; _(4.24)

(ii) forr=2, we have

A2∗A−2+A−2∗A2+ 2I≥A∗A−1+A−1∗A2. (4.25)

Theorem4.9. LetA1≥0,A2≥0be compatible partitioned and letIbe a compatible

par-titioned identity matrix. Then

A2

1∞A22+ 2(A1∗A2)≥

A1∞A2

2

, (4.26)

whereA1∞A2=A1∗I+I∗A2is called the Khatri-Rao sum.

Proof. SetL=A1∇A2=A1ΘI+IΘA2(Tracy-Singh sum). SinceA1≥0 andA2≥0, then A∗1 =A1andA∗2 =A2. Calculations show that

ZT

1LL∗Z1=Z1TLLZ1=Z1T

A1ΘI+IΘA2

A1ΘI+I2ΘA2

Z1

=A2

1∗I+I∗A22+ 2

A1∗A2

=A2

1∞A22+ 2

A1∗A2

. (4.27)

Similarly,

ZT

1LZ2

ZT

1LZ2

∗₌_ZT

1

A1ΘI+IΘA2

Z2

ZT

1

A1ΘI+IΘA2

Z2

∗

=A1∗I+I∗A2

2

=A1∞A2

2

. (4.28)

Substituting (4.27) and (4.28) into (3.2), we get (4.26).

Theorem4.10. LetA1>0andA2>0be compatible partitioned matrices. Then for any

positive real numberr,

rA2 1∗A22

+A1A2∗A2A1+A2A1∗A1A2+1 r

A2 2∗A21

≥rA1∗A2

2

+A1∗A2

A2∗A1

+A2∗A1

A1∗A2

+1

r

A2∗A1

2 .

(4.29)

Proof. SetL=ε1A1ΘA2+ε2A2ΘA1, whereε1andε2are both positive. SinceA1>0 and A2>0, thenA∗1 =A1andA∗2 =A2. Compute

ZT

1LL∗Z1=ZT1LLZ1=Z1T

ε1A1ΘA2+ε2A2ΘA1

Z1

=ZT

1

ε2 1

A2 1ΘA22

+ε1ε2

A1A2ΘA2A1

+ε1ε2

A2A1ΘA1A2

+ε2

2

A2 2ΘA21

Z1

=ε21

A21∗A22

+ε1ε2

A1A2∗A2A1

+ε1ε2

A2A1∗A1A2

+ε22

A22∗A21

.

(18)

Similarly,

ZT

1LZ2

ZT

1LZ2

∗₌_ZT

1

ε1A1ΘA2+ε2A2ΘA1

Z2 ZT 1

ε1A1ΘA2+ε2A2ΘA1

Z2

∗

=ZT

1

ε1A1ΘA2+ε2A2ΘA1

Z2 ZT 1

ε1A1ΘA2+ε2A2ΘA1

Z2

=ε1A1∗A2+ε2A2∗A1

2

=ε2 1

A1∗A2

2

+ε1ε2

A1∗A2

A2∗A1

+ε1ε2

A2∗A1

A1∗A2

+ε22

A2∗A1

2 .

(4.31)

Substituting (4.30) and (4.31) into (3.2), we have

ε21

A21∗A22

+ε1ε2

A1A2∗A2A1

+ε1ε2

A2A1∗A1A2

+ε22

A22∗A21

≥ε2 1

A∗1A2

2

+ε1ε2

A1∗A2

A2∗A1

+ε1ε2

A2∗A1

A1∗A2

+ε2

2

A∗2A1

2 .

(4.32)

Setr=ε1/ε2, we get (4.29).

Remark 4.11. LetAi (1≤i≤k, k≥2) be compatible partitioned matrices. Then (3.7) can be proved by settingT1=

k

i=1ΘIandT2=

k

i=1ΘAi. Calculations show that

T2T2∗=

k

i=1

ΘAiA∗i , T2T1∗=

k

i=1

ΘAi, T1T2∗=

_k

i=1

ΘAi ∗

, T1T1∗=

k

i=1

ΘI.

(4.33)

Substituting these into (4.4) and using (2.11), we get (3.7).

Remark 4.12. LetAi(1≤i≤2) be compatible partitioned matrices. Then (3.14) can be proved by puttingk=2 inRemark 4.11.

Remark 4.13. All results obtained in Sections3 and 4are quite general. These results lead to inequalities involving Hadamard product, as a special case, for nonpartitioned matricesAi (i=1, 2,. . .,k, k≥2) with the Hadamard product and Kronecker product replacing the Khatri-Rao product and Tracy-Singh product, respectively.

Now we utilize the commutativity of the Hadamard product to develop, for instance, (3.7) ofTheorem 3.2. This result leads to the following inequality involving Hadamard product, as a special case:

k

i=1

◦AiA∗i

≥

_k

i=1

◦Ai

k

i=1

◦Ai ∗

. (4.34)

It is possible to develop (4.34) in a diﬀerent direction from (3.6). For example, Visick [8, Theorem 11, page 54] proved that ifA1,A2∈Mm,nands∈[−1, 1], then

A1A∗1 ◦A2A∗2 +s

A1A∗2◦A2A∗1

≥(1 +s)A1◦A2

A1◦A2

(19)

We will extend this inequality to the case of products involving any finite number of matrices.

If the Tracy-Singh and Khatri-Rao products are replaced by the Kronecker and Hada-mard products inLemma 2.2, respectively, we obtain the following corollary.

Corollary4.14. LetAi∈Mm,n(1≤i≤k,k≥2). Then

k

i=1

◦Ai=PkmT

_k

i=1

⊗Ai

Pkn, (4.36)

wherePkm=(E(11m)0(m)··· 0(m)E (m)

22 0(m)···0(m)···0(m)···0(m)E (m)

mm)T is of ordermk×m,0(m)is an

m×mmatrix with all entries equal to zero, andEi j(m)is anm×mmatrix of zeros except for a one in the(i,j)th position.

Theorem4.15. LetAi∈Mm,n(1≤i≤k, k≥2). Then for any real scalars α1,α2,. . .,αk which are not all zero,

α2

1+···+α2k k

i=1

◦AiA∗i

+ _k₋₁

r=1 μr

k

w=1

◦AwA∗(w+r)

≥α1+···+αk 2

_k

i=1

◦Ai

_k

i=1

◦Ai ∗

,

(4.37)

whereμr= k

w=1αwα(w+r)andw+r≡(w+r)modkwith1≤(w+r)≤k. Proof. Let

L=α1A1⊗A2⊗ ··· ⊗Ak+α2A2⊗ ··· ⊗Ak⊗A1+···+αkAk⊗A1⊗ ··· ⊗Ak−1,

(4.38)

whereAi∈Mm,n(1≤i≤k,k≥2) andα1,α2,. . .,αkare real scalars which are not all zero. Taking indices “ modk,”Lemma 2.1(i), (iii) (by setting⊗instead ofΘ) give

LL∗= k

i=1

αiAi⊗Ai+1⊗ ··· ⊗Ai−1

k

i=1

αiA∗i ⊗A∗i+1⊗ ··· ⊗A∗i−1

=α2 1

A1A∗1⊗ ··· ⊗AkA∗k

+···+α2

k

AkA∗k ⊗AA1∗⊗ ··· ⊗AkA∗k−1

+

i=j

αiαj

AiA∗j ⊗Aj+1A∗j+1⊗ ··· ⊗Aj−1A∗j−1

.

(4.39)

Now the application of (4.36) and the commutativity of the Hadamard product yield

PT_kmLL∗Pkm=α21+···+α2k k

i=1

◦AiA∗i

+ _k₋₁

r=1 μr

k

w=1

◦AwA∗(w+r)

, (4.40)

(20)

Also by (4.36) and the commutativity of the Hadamard product, we obtain

P_kmT LPkn

=P_kmT α1

A1⊗A2⊗ ··· ⊗Ak

+α2

A2⊗ ··· ⊗Ak⊗A1

+···+αk

Ak⊗A1⊗ ··· ⊗Ak−1

Pkn

=α1PTkm

A1⊗A2⊗ ··· ⊗AkPkn+α2PTkm(A2⊗ ··· ⊗Ak⊗A1

Pkn

+···+αkPTkm

Ak⊗A1⊗ ··· ⊗Ak−1

Pkn

=α1

A1◦A2◦ ··· ◦Ak

+α2

A2◦ ··· ◦Ak◦A1

+···+αkAk◦A1◦ ··· ◦Ak−1

=α1+···+αk k

i=1

◦Ai

,

P_kmT LPkn ∗

=α1+···+αk k

i=1

◦Ai ∗

.

(4.41)

Now

PT kmLPkn

∗₌_α

1+···+αk 2

_k

i=1

◦Ai

_k

i=1

◦Ai ∗

. (4.42)

SinceP_kmT LL∗Pkm≥(PkmT LPkn)(PkmT LPkn)∗ by (3.2) and from (4.40) and (4.42), we get

(4.37).

Now, we examine some special cases briefly.

In order to see that (4.37) really is an extension in (4.34), it is suﬃcient to setα1=1

andα2= ··· =αk=0. Thus we recover the result of Visick in (4.35) which we mentioned before the statement ofCorollary 4.14. Letk=2, thenμ1=

2

w=1αwα(w+1)withw+ 1≡ (w+ 1)mod 2, that is,μ1=2α1α2. ThenTheorem 4.15asserts that

α21+α22

A1A∗1 ◦A2A∗2

+ 2α1α2

A1A∗2◦A2A∗1

≥α1+α2

2 A1◦A2

A1◦A2

∗

.

(4.43)

Simplification gives

A1A∗1 ◦A2A∗2 +s

A1A∗2◦A2A∗1

≥(1 +s)A1◦A2

A1◦A2

∗

(4.44)

for anys∈[−1, 1], just as we wanted. Finally, we present an attractive inequality using three matrices. Letk=3,α1=1,α2=α3= −1/2.Theorem 4.15asserts that

A1A∗1 ◦A2A∗2 ◦A3A∗3 ≥

1 2

A1A∗2◦A2A∗3◦A3A∗1+A2A∗1 ◦A3A∗2 ◦A1A∗3

(21)

5. Acknowledgments

The authors would like to thank the referees for their valuable comments and sugges-tions, including the simplified proof ofTheorem 3.6and some statements. The present research has been partially supported by University Putra Malaysia (UPM) under the Grant IRPA09-02-04-0259-EA001.

References

[1] Z. A. Al Zhour and A. Kilicman,New Holder-type inequalities for the Tracy-Singh and Khatri-Rao products of positive matrices, Proceedings of the International Conference on Mathematics, Statistics and Their Applications, vol. 1, North Sumatera, 2005, pp. 1–7.

[2] A. Albert,Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM Journal on Applied Mathematics17(1969), no. 2, 434–440.

[3] C.-G. Cao, X. Zhang, and Z.-P. Yang,Some inequalities for the Khatri-Rao product of matrices, Electronic Journal of Linear Algebra9(2002), 276–281.

[4] J. Chollet,Some inequalities for principal submatrices, The American Mathematical Monthly104 (1997), no. 7, 609–617.

[5] S. Liu,Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra and Its Appli-cations289(1999), no. 1–3, 267–277.

[6] ,Several inequalities involving Khatri-Rao products of positive semidefinite matrices, Lin-ear Algebra and Its Applications354(2002), no. 1–3, 175–186.

[7] B. Mond and J. E. Peˇcari´c,Matrix inequalities for convex functions, Journal of Mathematical Anal-ysis and Applications209(1997), no. 1, 147–153.

[8] G. Visick,A quantitative version of the observation that the Hadamard product is a principal sub-matrix of the Kronecker product, Linear Algebra and Its Applications304(2000), no. 1–3, 45–68. [9] F. Zhang,Matrix Theory. Basic Results and Techniques, Universitext, Springer, New York, 1999. [10] X. Zhang, Z.-P. Yang, and C.-G. Cao,Inequalities involving Khatri-Rao products of positive

semi-definite matrices, Applied Mathematics E-Notes2(2002), 117–124.

Zeyad Abdel Aziz Al Zhour: Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

E-mail address:[email protected]

Adem Kilicman: Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia