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INVOLVING THE KHATRI-RAO PRODUCT OFSEVERAL POSITIVE MATRICES ZEYAD ABDEL AZIZ AL ZHOUR AND ADEM KILICMAN Received 15 February 2005; Accepted 16 October 2005 Recently, there have been man

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 volving Khatri-Rao and Hadamard products of two positive matrices In this paper, theresults are established in the following three ways First, we find generalization of theinequalities involving Khatri-Rao product using results given by Liu (1999), Mond andPeˇcari´c (1997), Cao et al (2002), Chollet (1997), and Visick (2000) Second, we recoverand 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 involvingHadamard product, as a special case

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

distrib-1 Introduction

Consider matricesA and B of order m × n and p × q, respectively Let A =[A i j] be titioned with A i j of order m i × n j as the (i, j)th block submatrix and let B =[B kl] bepartitioned withB kl of order p k × q l as the (k, l)th block submatrix (m =t

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 =a i j B

whereA =[a i j],B =[b kl] are scalar matrices of orderm × n and p × q,

respec-tively,a i j B is of order p × q, and A ⊗ B of order mp × nq;

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 80878, Pages 1 21

DOI 10.1155/JIA/2006/80878

(ii) Hadamard product

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

q, respectively, A i j is of orderm i × n j,B kl of order p k × q l,A i j ⊗ B kl of order

whereA =[A i j],B =[B i j] are partitioned matrices of orderm × n and p × q,

respectively,A i jis of orderm i × n j,B klof orderp i × q j,A i j ⊗ B i jof orderm i p i ×

(1) for a nonpartitioned matrixA, their A ΘB is A ⊗ B, that is,

of zeros and ones such thatZ T1Z1= I r,Z2T Z2= I s (I r andI sarer × r and s × s identity

Z t

⎤

⎥

where eachZ i =[0i1 ···0ii −1I mi pi 0i i+1 ···0it]T is an real matrix of zeros and ones, and 0ik is

am i p i × m i p kzero matrix for anyk = i Note also that Z T

Khatri-We use the following notations:

(i)M m,n—the set of allm × n matrices over the complex number fieldCand when

m = n, we write M minstead ofM m,n;

(ii)A T,A ∗,A+,A −1—the transpose, conjugate transpose, Moore-Penrose inverse,and inverse of matrixA, respectively.

For Hermitian matricesA and B, the relation A > B means that A − B > 0 is a positive

definite and the relationA ≥ B means A− B ≥0 is a positive semidefinite Given a positivedefinite matrixA, its positive definite square root is denoted by A1/2 We use the knownfact “for positive definite matricesA and B, the relation A ≥ B implies A1/2 ≥ B1/2” which

is called the L¨owner-Heinz theorem.

2 Some notations and preliminary results

LetA be a positive definite m × m matrix The spectral decomposition of matrix A assures

that there exists a unitary matrixU such that

A = U ∗ DU = U ∗diag

λ i

whereD =diag(λ i)=diag(λ1, , λ m) is the diagonal matrix with diagonal entriesλ i(λ i

are the positive eigenvalues ofA) For any real number r, A ris defined by

A r = U ∗ D r U = U ∗diag

λ r i

IfA ∈ M m,nis any matrix with rank (A)= s, the singular value decomposition of A assures

that there are unitary matricesU ∈ M mandV ∈ M nsuch that

Here

=[W 00 0]∈ M m,n, whereW =diag(σ1, , σ s)∈ M sis the diagonal matrix with agonal entriesσ i (i =1, 2, , s) and σ1≥ σ2≥ ··· ≥ σ s > 0 are the singular values of A,

di-that is,σ1≥ σ2≥ ··· ≥ σ s > 0 are positive square roots of positive eigenvalues of A ∗ A and

AA ∗ The Moore-Penrose inverse of A is defined by

For any compatible partitioned matricesA, B, C, and D, we will make a frequent use

of the following properties of the Tracy-Singh product (see e.g., [1,3,5,10]):

(a) (A ΘB)(CΘD) =(AC) Θ(BD) if AC and BD are well defined;

(b) (A ΘB) r = A r ΘB r ifA ∈ M m, B ∈ M nare positive semidefinite matrices andr is

any real number;

whereλ1(A), λ m(A) are the largest and smallest eigenvalues, respectively, of a matrix A,

andλ1(B), λ n(B) are the largest and smallest eigenvalues, respectively, of a matrix B.

The Khatri-Rao and Tracy-Singh products ofk matrices A i(1≤ i ≤ k, k ≥2) will bedenoted byk

i =1∗A i = A1∗ A2∗ ··· ∗ A k and k

i =1ΘA i = A1ΘA2Θ··· ΘA k, tively

respec-For a finite number of matricesA i (i =1, 2, , k), the properties (a)–(d) become as

inLemma 2.1and the connection between the Khatri-Rao and Tracy-Singh products in(1.7) and (1.8) becomes as inLemma 2.2

Lemma 2.1 Let A i and B i(1≤ i ≤ k, k ≥ 2) be compatible partitioned matrices Then

k

i =1

B i

, k =2, 3, . (2.10)

Proof The proof is immediately derived by induction on k. 

Lemma 2.2 Let A i =[A(gh i)]∈ M m(i),n(i) (1≤ i ≤ k, k ≥ 2) be partitioned matrices with

A(gh i) as the (g, h)th block submatrix (m =k

m(i) = n(i) (1 ≤ i ≤ k, k ≥ 2), then there exists an m × r matrix Z of zeros and ones such that Z T Z = I r ,

matrices, that is, there exist anm × r matrix P krof zeros and ones and ann × s matrix R ks

of zeros and ones such thatP kr T P kr = I r,R T ks R ks = I s, and



P T kr

LettingZ1=(I m(1) ΘP kr)Q1andZ2=(I n(1) ΘR ks)Q2, the inductive step is complete Here

Q1= P2r = P r,Q1= R2s = R s, and it is a simple matter to verify that

Z1=I m(1) ΘP kr

P r = P(k+1)r, Z1T = P r T

I m(1) ΘP T kr

I n(1) ΘR T ks

= R T

(k+1)s (2.16)

Lemma 2.4 Let X j > 0 ( j =1, 2, , k) be n × n matrices with eigenvalues in the interval

[w, W] and U j (j =1, 2, , k) are r × m matrices such that k

j =1U j U ∗ j = I Then (see Mond and Peˇcari´c [ 7 ])

(i) for every real p > 1 and p < 0,

While for 0 < p < 1, the reverse inequality holds in ( 2.20 );

(ii) for every real p > 1 and p < 0,

3 New applications and results

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

Theorem 3.1 Let A i =[A(gh i)]∈ M m(i),n(i)(1≤ i ≤ k, k ≥ 2) be partitioned matrices with

A(gh i) as the (g, h)th block submatrix (m =k

i =1m(i), n =k

i =1n(i)) and let Z1and Z2be the real matrices of zeros and ones that satisfy ( 2.11 ) Then

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

Ω=[Z1Q(m) ] is an m × m permutation matrix Q(m) is not unique but for any such choice of Q(m) ,

(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

Theorem 3.2 Let A i =[A(gh i)]∈ M m,n(1≤ i ≤ k, k ≥ 2) be partitioned matrices with A(gh i)

as the (g, h)th block submatrix Let Z1be an m k × r matrix of zeros and ones that satisfies ( 2.12 ) and let Q(n) be an n k ×(n k − s) matrix of zeros and ones that satisfies ( 3.1 ) Then

Z2Z T

2 k

to (2.11) ofLemma 2.2there exist two real matricesZ1andZ2of zeros and ones of order

m k × r and n k × s, respectively, such that

But becauseA i A ∗ 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 orderm k × r such

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

If we putk =2 inTheorem 3.2, we obtain the following corollary

Corollary 3.3 Let A i =[A(gh i)]∈ M m,n(1≤ i ≤ 2) be partitioned matrices with A(gh i) as the

(g, h)th block submatrix Let Z1be an m2× r matrix of zeros and ones that satisfies ( 1.8 ) and let Q(n) be an n2×(n2− s) matrix of zeros and ones that satisfies ( 3.1 ) Then

A1A ∗1 ∗ A2A ∗2 =A1∗ A2

A1∗ A2

∗+Z T

Corollary 3.4 Let A i =[A(gh i)]∈ M m,n(1≤ i ≤ k, k ≥ 2) be partitioned matrices with A(gh i)

as the (g, h)th block submatrix Let Z1be an m k × r matrix of zeros and ones that satisfies ( 2.12 ) and let Q(n) be an n k ×(n k − s) matrix of zeros and ones that satisfies ( 3.1 ) Then the following statements are equivalent:

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 =1ΘA i)Q(n) =0 To arrive from (ii) to (iii), noticethat (ii) may be rewritten asZ1T(k

i =1ΘA i)Q(n) Q(T n) =0 ByTheorem 3.1(i), there exist an

n k × s matrix Z2of zeros and ones that satisfies (2.12) and ann k ×(n k − s) matrix Q(n)ofzeros and ones that satisfies (3.1) such thatQ(n) Q T(n) = I n k − Z2Z2T, this becomes

If we putk =2 inCorollary 3.4, we obtain the following corollary

Corollary 3.5 Let A i =[A(gh i)]∈ M m,n(1≤ i ≤ 2) be partitioned matrices with A(gh i) as the

(g, h)th block submatrix Let Z1be an m2× r matrix of zeros and ones that satisfies ( 1.8 ) and let Q(n) be an n2×(n2− s) matrix of zeros and ones that satisfies ( 3.1 ) Then the following statements are equivalent:

Theorem 3.6 Let A i ≥0 (1≤ i ≤ k, k ≥ 2) be n × n compatible partitioned matrices Then (i) if either −1≤ r ≤ 0 or 1 ≤ r ≤ 2, then

inLemma 2.3and applying (2.12) ofLemma 2.2.

Theorem 3.8 Let A i > 0 be compatible partitioned matrices such thatk

i =1ΘA i > 0 (1 ≤

i ≤ k, k ≥ 2) Let W and w be the largest and smallest eigenvalues ofk

i =1ΘA i , respectively Then

(i) for every real p > 1 and p < 0,

While for every 0 < p < 1, the reverse inequality holds in ( 3.25 );

(ii) for every real p > 1 and p < 0,

While for every 0 < 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 thesake of convenience In (2.20) and (2.22) ofLemma 2.4, setk =1 and replaceU by Z T,

U ∗ byZ, and X by k

i =1ΘA i, 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

(ii) forp = −1, we have

From (3.27), we have the following special cases:

(i) forp =2, we have



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

4 Further developments and applications

Due to Albert’s theorem in [2] and [9, Theorem 6.13], for a partitioned matrix [B A B ∗ D] with

a positive (semi) definite matrixA ∈ M m,

for any positive semidefinite matrixD ∈ M n It is also known that if matrixA is square

and nonsingular, thenA+= A −1and [B A B ∗ D]≥0 if and only ifD ≥ B ∗ A −1B

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

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

Therefore (4.4) can be considered to be more general than (3.2) In order to prove this wesetT1= I and T2= L in (4.4), we have

56, Theorems 11, 17, and 20] and establish some new inequalities involving Khatri-Raoproducts of several positive matrices

Theorem 4.1 Let A1and A2be compatible partitioned matrices Then

A2∗ A1

∗

Proof Set T1= I ΘI and T2 = A1ΘA2+A2ΘA1 Then calculations show that

T2T2∗ = A1A ∗1ΘA2 A ∗2 +A2A ∗2ΘA1 A ∗1 +A1A ∗2ΘA2 A ∗1+A2A ∗1ΘA1 A ∗2,

T2T1∗ = A1ΘA2+A2ΘA1, T1T2∗ =A1ΘA2∗+

A2ΘA1∗, T1T1∗ = I ΘI.

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

Corollary 4.2 Let A i(1≤ i ≤ 2) be Hermitian compatible partitioned matrices Then

I ∗ A2≥I ∗ A2

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

(ii) SetA1= A and A2= A −1in (4.8), we get (4.9)

(iii) SetA1= I and A2= A in (4.8), we get (4.10) 

Corollary 4.3 Let A i > 0 (1 ≤ i ≤ 2) be compatible partitioned matrices Then

A2∗ A2 1/2

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

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

If we putk =2 and replaceA ibyA r

i(1≤ i ≤2) inTheorem 4.4, we obtain the followingtheorem

Theorem 4.5 Let A1≥ 0, A2≥ 0 be compatible partitioned and let r be any nonzero real number such that A0= A r/21 A+1r/2 = A+1r/2 A r/21 and A0= A r/22 A+2r/2 = A+2r/2 A r/22 Then

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

Theorem 4.6 Let A1> 0, A2> 0 be compatible partitioned and let I be a compatible tioned identity matrix Then for any nonzero real number r,

If we putr =1 andA1= A2inTheorem 4.6, we obtain the following theorem

Theorem 4.7 Let A > 0 be compatible partitioned and let I be a compatible partitioned identity matrix Then

2I + A ∗ A −1+A −1∗ A ≥(A ∗ I + I ∗ A)(A ∗ A) −1(A ∗ I + I ∗ A). (4.16)

In particular, if I is a nonpartitioned identity matrix, then

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

From (4.19), we have the following special cases:

(i) forr =1, we have

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

Proof Set L = A1∇A2= A1ΘI + IΘA2(Tracy-Singh sum) Since A1≥0 andA2≥0, then

A ∗1 = A1andA ∗2 = A2 Calculations show that

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

Theorem 4.10 Let A1> 0 and A2> 0 be compatible partitioned matrices Then for any positive real number r,

A1∗ A2

A2∗ A1

+

A2∗ A1

A1∗ A2

+1

Proof Set L = ε1A1ΘA2+ε2A2ΘA1, whereε1andε2are both positive SinceA1> 0 and

A2> 0, then A ∗1 = A1andA ∗2 = A2 Compute

A2A1ΘA1A2

+ε2

A2A1∗ A1A2

+ε2

A2∗ A2

.

(4.30)

A1∗ A2

A2∗ A1 +ε1ε2

A2∗ A1

2

.

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

A2A1∗ A1A2

+ε22

A22∗ A21

≥ε2

A ∗1A2

2+ε1ε2

Remark 4.12 Let A i(1≤ i ≤2) be compatible partitioned matrices Then (3.14) can beproved by puttingk =2 inRemark 4.11

Remark 4.13 All results obtained in Sections3 and 4are quite general These resultslead to inequalities involving Hadamard product, as a special case, for nonpartitionedmatricesA i (i =1, 2, , k, k ≥2) with the Hadamard product and Kronecker productreplacing 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 Hadamardproduct, as a special case:

. (4.35)

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

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

Hada-Corollary 4.14 Let A i ∈ M m,n(1≤ i ≤ k, k ≥ 2) Then

m × m matrix with all entries equal to zero, and E i j(m) is an m × m matrix of zeros except for

a one in the (i, j)th position.

Theorem 4.15 Let A i ∈ M m,n(1≤ i ≤ k, k ≥ 2) Then for any real scalars α1,α2, , α k which are not all zero,

Now we utilize the commutativity of the Hadamard product to develop, for instance,(3.7) ofTheorem... have

56, Theorems 11, 17, and 20] and establish some new inequalities involving Khatri-Raoproducts of several positive matrices

Theorem 4.1 Let A1and A2be... Chollet [4], and Mond andPeˇcari´c [7] with respect to the Khatri-Rao product and extend these results to any finitenumber of matrices These results lead to inequalities involving Hadamard products,