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EXTENSION AND GENERALIZATION INEQUALITIES
INVOLVING THE KHATRI-RAO PRODUCT OF
SEVERAL POSITIVE MATRICES
ZEYAD ABDEL AZIZ AL ZHOUR AND ADEM KILICMAN
Received 15 February 2005; Accepted 16 Oc tober 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
Pe
ˇ
cari
´
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 matrices A and B of order m
×n and p ×q, respectively. Let A = [A
ij
]bepar-
titioned with A
ij
of order m
i
×n

j
as the (i, j)th block submatrix and let B = [B
kl
]be
partitioned with B
kl
of order p
k
×q
l
as the (k,l)th block submatrix (m =

t
i
=1
m
i
, n =

d
j
=1
n
j
, p =

u
k
=1
p

k
, q =

v
l
=1
q
l
). For simplicity, we say that A and B are compatible
partitioned if A
= [A
ij
]
t
i, j
=1
and B = [B
ij
]
t
i, j
=1
are square matrices of order m ×m and
partitioned, respectively, with A
ij
and B
ij
of order m
i
×m

j
(m =

t
i
=1
m
i
=

t
j
=1
m
j
).
Let A
⊗B, A ◦B, AΘB,andA ∗B be the Kronecker, Hadamard, Tracy-Singh, and
Khatri-Rao products, respectively, of A and B. The definitions of the mentioned four
matrix products are given by Liu in [5, 6]asfollows:
(i) Kronecker product
A
⊗B =

a
ij
B

ij
, (1.1)

where A
= [a
ij
], B = [b
kl
] are scalar matrices of order m ×n and p ×q,respec-
tively, a
ij
B is of order p ×q,andA ⊗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
2 Generalization inequalities for Khatri-Rao product
(ii) Hadamard product
A
◦B =

a
ij
b
ij

ij
= B ◦A, (1.2)
where A
= [a
ij
], B =[b
ij

] are scalar matrices of order m ×n, a
ij
b
ij
is a scalar, and
A
◦B is of order m ×n;
(iii) Tracy-Singh product
AΘB
=

A
ij
ΘB

ij
=

A
ij
⊗B
kl

kl

ij
, (1.3)
where A
= [A
ij

], B = [B
kl
] are partitioned matrices of order m ×n and p ×
q, respectively, A
ij
is of order m
i
×n
j
, B
kl
of order p
k
×q
l
, A
ij
⊗B
kl
of order
m
i
p
k
×n
j
q
l
, A
ij

ΘB of order m
i
p ×n
j
q (m =

t
i
=1
m
i
, n =

d
j
=1
n
j
, p =

u
k
=1
p
k
,
q
=

v

l
=1
q
l
), and AΘB of order mp×nq;
(iv) Khatri-Rao product
A
∗B =

A
ij
⊗B
ij

ij
, (1.4)
where A
= [A
ij
], B = [B
ij
] are partitioned matrices of order m ×n and p ×q,
respectively, A
ij
is of order m
i
×n
j
, B
kl

of order p
i
×q
j
, A
ij
⊗B
ij
of order m
i
p
i
×
n
j
q
j
(m =

t
i
=1
m
i
, n =

d
j
=1
n

j
, p =

t
i
=1
p
i
, q =

d
j
=1
q
j
), and A ∗B of order
M
×N (M =

t
i
=1
m
i
p
i
,N =

d
j

=1
n
j
q
j
).
In general, AΘB
= BΘA, A ⊗B = B ⊗A, A ∗B = B ∗A,butifA =[a
ij
]isascalarmatrix
and B
= [B
ij
] is a partitioned matrix, then A ∗ 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 matr i x A, their AΘB is A
⊗B, that is,
AΘB
=

a
ij
ΘB

ij
=

a
ij

⊗B
kl

kl

ij
=

a
ij
B
kl

kl

ij
=

a
ij
B

ij
= A⊗B; (1.5)
(2) for nonpartitioned matrices A and B of order m
×n, their A ∗B is A ◦B, that is,
A
∗B =

a

ij
⊗b
ij

ij
=

a
ij
b
ij

ij
= A◦B. (1.6)
The Khatri-Rao and Tracy-Singh products are related by the following relation [5, 6]:
A
∗B =Z
T
1
(AΘB)Z
2
, (1.7)
where A
= [A
ij
]ispartitionedwithA
ij
of order m
i
×n

j
and B =[B
kl
]ispartitionedwith
B
kl
of order p
k
×q
l
(m =

t
i
=1
m
i
, n =

d
j
=1
n
j
, p =

u
k
=1
p

k
, q =

v
l
=1
q
l
), Z
1
is an mp×
r (r =

t
i
=1
m
i
p
i
) mat rix of zeros and ones, and Z
2
is an nq × s (s =

d
j
=1
n
j
q

j
)matrix
Z. A. Al Zhour and A. Kilicman 3
of zeros and ones such that Z
T
1
Z
1
= I
r
, Z
T
2
Z
2
= I
s
(I
r
and I
s
are r × r and s ×s identity
matrices, resp.).
In particular, if m
= n and p = q, then there exists an mp×r (r =

t
i
=1
m

i
p
i
)matrixZ
such that Z
T
Z =I
r
(I
r
is an r ×r identity matrix) and
A
∗B =Z
T
(AΘB)Z. (1.8)
Here
Z
=
⎡
⎢
⎢
⎣
Z
1
.
.
.
Z
t
⎤

⎥
⎥
⎦
, (1.9)
where each Z
i
= [
0
i1
··· 0
ii−1
I
m
i
p
i
0
ii+1
··· 0
it
]
T
is an real matrix of zeros and ones, and 0
ik
is
a m
i
p
i
×m

i
p
k
zero matrix for any k = i.NotealsothatZ
T
i
Z
i
= I and
Z
T
i

A
ij
ΘB

Z
j
= Z
T
i

A
ij
⊗B
kl

kl
Z

j
= A
ij
⊗B
ij
, 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) M
m,n
—the set of all m ×n matrices over the complex number field C and when
m
= n,wewriteM
m
instead of M
m,n
;
(ii) A
T
,A
∗
,A
+
,A

−1
—the tr anspose, conjugate t ranspose, Moore-Penrose inverse,
and inverse of matrix A, respectively.
For Hermitian mat rices A and B,therelationA>Bmeans that A
−B>0 is a positive
definite and the relation A
≥ B means A−B ≥0 is a positive semidefinite. Given a positive
definite matrix A, its positive definite square root is denoted by A
1/2
. We use the known
fact “for positive definite matrices A and B,therelationA
≥ B implies A
1/2
≥ B
1/2
” which
is called the L
¨
owner-Heinz theorem.
2. Some notations and preliminary results
Let A be a positive definite m
×m matrix. The spectral decomposition of matrix A assures
that there exists a unitary matrix U such that
A
= U
∗
DU =U
∗
diag


λ
i

U, U
∗
U =I
m
, (2.1)
4 Generalization inequalities for Khatri-Rao product
where D
= diag(λ
i
) = diag(λ
1
, ,λ
m
) is the diagonal matrix with diagonal entries λ
i
(λ
i
are the positive eigenvalues of A). For any real number r, A
r
is defined by
A
r
= U
∗
D
r
U =U

∗
diag

λ
r
i

U. (2.2)
If A
∈ M
m,n
is any matrix with rank (A) =s,thesingular value decomposition of A assures
that there are unitary matrices U
∈ M
m
and V ∈ M
n
such that
A
= U

V
∗
. (2.3)
Here

=
[
W 0
00

] ∈M
m,n
,whereW = diag(σ
1
, ,σ
s
) ∈M
s
is the diagonal matrix with di-
agonal entries σ
i
(i = 1,2, ,s)andσ
1
≥ σ
2
≥···≥σ
s
> 0 are the singular values of A,
that is, σ
1
≥ σ
2
≥···≥σ
s
> 0 are positive square roots of positive eigenvalues of A
∗
A and
AA
∗
.TheMoore-Penrose inverse of A is defined by

A
+
= V

W
−1
0
00

U
∗
∈ M
n,m
, (2.4)
where W
−1
= diag(σ
−1
1
,σ
−1
2
, ,σ
−1
s
) ∈ M
s
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 matrices A, 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)ifAC and BD are well defined;
(b) (AΘB)
r
= A
r
ΘB
r
if A ∈ M
m
, B ∈ M
n
are positive semidefinite matrices and r is
any real number;
(c) (AΘB)
∗
= A
∗
ΘB
∗
;
(d) (AΘB)
+
= A
+
ΘB
+
.

If A
∈ M
m
and B ∈M
n
are positive semidefinite matrices, then (see, [3, 10])
(e) AΘB
≥ 0;
(f) λ
1
(AΘB) =λ
1
(A)λ
1
(B), λ
mn
(AΘB) =λ
m
(A)λ
n
(B),
where λ
1
(A), λ
m
(A) are the largest and smallest eigenvalues, respectively, of a matrix A,
and λ
1
(B), λ
n

(B) are the largest and smallest eigenvalues, respectively, of a matr ix B.
The Khatri-Rao and Tracy-Singh products of k matrices A
i
(1 ≤ i ≤k, k ≥2) will be
denoted by

k
i
=1
∗A
i
= A
1
∗A
2
∗···∗A
k
and

k
i
=1
ΘA
i
= A
1
ΘA
2
Θ ···ΘA
k

,respec-
tively.
For a finite number of matrices A
i
(i = 1,2, ,k), the properties (a)–(d) become as
in Lemma 2.1 and the connection between the Khatr i-Rao and Tracy-Singh products in
(1.7)and(1.8)becomesasinLemma 2.2.
Z. A. Al Zhour and A. Kilicman 5
Lemma 2.1. Let A
i
and B
i
(1 ≤i ≤k, k ≥2) be compatible partitioned matrices. Then
(i)

k

i=1
ΘA
i

k

i=1
ΘB
i

=

k


i=1
Θ

A
i
B
i


(2.6)
if A
i
B
i
(1 ≤i ≤k, k ≥2) are well defined;
(ii)

k

i=1
ΘA
i

+
=
k

i=1
ΘA

+
i
, k =2,3, ; (2.7)
(iii)

k

i=1
ΘA
i

∗
=
k

i=1
ΘA
∗
i
,

k

i=1
∗A
i

∗
=
k


i=1
∗A
∗
i
, k = 2,3, ; (2.8)
(iv)

k

i=1
ΘA
i

r
=
k

i=1
ΘA
r
i
if A
i
∈ M
m(i)
(1 ≤i ≤k, k ≥2) (2.9)
are positive semidefinite matrices and r is any real number;
(v)


k

i=1

A
i
ΘB
i


=

k

i=1
A
i

Θ

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
(i)
gh
] ∈ M
m(i),n(i)
(1 ≤ i ≤ k, k ≥ 2) be partitioned matrices with
A
(i)
gh
as the (g,h)th block submatrix (m =

k
i
=1
m(i), n =

k
i
=1
n(i), r =

t
j
=1

k
i
=1
m

j
(i),
s
=

t
j
=1

k
i
=1
n
j
(i), m(i) =

t
j
=1
m
j
(i), n(i) =

t
j
=1
n
j
(i)). Then there exist two real ma-
trices Z

1
of order m ×r and Z
2
of order n ×s such that Z
T
1
Z
1
= I
r
, Z
T
2
Z
2
= I
s
(Z
1
, Z
2
are real
matrices of zeros and ones) and
k

i=1
∗A
i
= Z
T

1

k

i=1
ΘA
i

Z
2
, k =2,3, , (2.11)
where I
r
and I
s
are identity matrices of order r ×r and s ×s, respectively. In particular, if
6 Generalization inequalities for Khatri-Rao product
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
,
k

i=1
∗A
i
= Z

T

k

i=1
ΘA
i

Z, k = 2,3, , (2.12)
and ZZ
T
is an m ×m diagonal matrix of zeros and ones, so
0
≤ ZZ
T
≤ I
m
, (2.13)
where m
=

k
i
=1
m(i).
Proof. The special case in (2.12)ofLemma 2.2 is proved in [3, Corollary 2.2] and (2.13)
follows immediately by the definition of matrix Z. We give proof of the general case in
(2.11)ofLemma 2.2 for the sake of convenience. We proceed by induction on k.Ifk
= 2,
then (2.11)istrueby(1.7). Now suppose (2.11) holds for the Khatri-Rao product of k

matrices, that is, there exist an m
×r matrix P
kr
of zeros and ones and an n ×s matr ix R
ks
of zeros and ones such that P
T
kr
P
kr
= I
r
, R
T
ks
R
ks
= I
s
,and
k

i=1
∗A
i
= P
T
kr

k


i=1
ΘA
i

R
ks
, k = 2,3, (2.14)
WewillprovethatitistruefortheKhatri-Raoproductofk + 1 matrices. Then by (1.7),
there exist an m(1)r
×r matrix Q
1
of zeros and ones and an n(1)s ×s matrix Q
2
of zeros
and ones such that Q
T
1
Q
1
= I
r
, Q
T
2
Q
2
= I
s
,and

k+1

i=1
∗A
i
= A
1
∗

k+1

i=2
∗A
i

=
Q
T
1

A
1
Θ
k+1

i=2
∗A
i

Q

2
= Q
T
1

A
1
Θ

P
T
kr

k+1

i=2
ΘA
i

R
ks

Q
2
= Q
T
1

I
m(1)

A
1
I
n(1)

Θ

P
T
kr

k+1

i=2
ΘA
i

R
ks

Q
2
= Q
T
1

I
m(1)
ΘP
T

kr


A
1
Θ

k+1

i=2
ΘA
i


I
n(1)
ΘR
ks

Q
2
= Q
T
1

I
m(1)
ΘP
T
kr



k+1

i=1
ΘA
i


I
n(1)
ΘR
ks

Q
2
.
(2.15)
Letting Z
1
= (I
m(1)
ΘP
kr
)Q
1
and Z
2
= (I
n(1)

ΘR
ks
)Q
2
, the inductive step is complete. Here
Q
1
= P
2r
= P
r
, Q
1
= R
2s
= R
s
, and it is a simple matter to verify that
Z
1
=

I
m(1)
ΘP
kr

P
r
= P

(k+1)r
, Z
T
1
= P
T
r

I
m(1)
ΘP
T
kr

=
P
T
(k+1)r
,
Z
2
=

I
n(1)
ΘR
ks

R
s

= R
(k+1)s
, Z
T
2
= R
T
s

I
n(1)
ΘR
T
ks

=
R
T
(k+1)s
.
(2.16)
Z. A. Al Zhour and A. Kilicman 7
Note that
Z
T
1
Z
1
= P
T

r

I
m(1)
ΘP
T
kr

I
m(1)
ΘP
kr

P
r
= Q
T
1

I
m(1)
ΘP
T
kr

I
m(1)
ΘP
kr


Q
1
= Q
T
1

I
m(1)
I
m(1)
ΘP
T
kr
P
kr

Q
1
= Q
T
1

I
m(1)
ΘI
r

Q
1


I
m(1)
ΘI
r
= I
m(1)r

=
Q
T
1

I
m(1)r

Q
1
= Q
T
1
Q
1
= I
r
.
(2.17)
Similarly, it is easy to verify that Z
T
2
Z

2
= I
s
. 
Lemma 2.3. Let α be a nonempty subset of the set {1,2, ,m} and let A ∈M
m
be a positive
semidefinite matrix. Then (see Chollet [4])
(i) if either
−1 ≤r ≤0 or 1 ≤r ≤2, then
A
r
(α) ≥A(α)
r
, ∀α; (2.18)
(ii) if 0
≤ r ≤ 1, then
A
r
(α) ≤A(α)
r
, ∀α, (2.19)
where A(α) is the principal submatrix of A whose entries are in the intersection of the rows
and columns of A specified by α.
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 matricessuchthat


k
j
=1
U
j
U
∗
j
= I. Then (see
Mond and Pe
ˇ
cari
´
c[7])
(i) for every real p>1 and p<0,
k

j=1
U
j
X
p
j
U
∗
j
≤ μ

k


j=1
U
j
X
j
U
∗
j

p
, (2.20)
where
μ
=
δ
p
−δ
(p −1)(δ −1)

p −1
p
δ
p
−1
δ
p
−δ

p

, δ =
W
w
. (2.21)
While for 0 <p<1, the reverse inequality holds in (2.20);
(ii) for every real p>1 and p<0,

k

j=1
U
j
X
p
j
U
∗
j

−

k

j=1
U
j
X
j
U
∗

j

p
≤ γ{I}, (2.22)
where
γ
=
Ww
p
−wW
p
W −w
+(p
−1)

1
p
W
p
−w
P
W −w

p/(p−1)
. (2.23)
While for 0 <p<1 , the reverse inequality holds in (2.22).
8 Generalization inequalities for Khatri-Rao product
3. New applications and results
Based on the basic results in Section 2 and the general connection between the Khatri-Rao
and Tracy-Singh products in Lemma 2.2, we generalize and derive some equalities and

inequalities in works of Visick [8, Corollary 3, Theorem 4], Chollet [4], and Mond and
Pe
ˇ
cari
´
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.
Theorem 3.1. Let A
i
= [A
(i)
gh
] ∈ M
m(i),n(i)
(1 ≤ i ≤k, k ≥2) be partitioned matrices with
A
(i)
gh
as the (g,h)th block submatrix (m =

k
i
=1
m(i), n =

k
i
=1
n(i)) and let Z

1
and Z
2
be 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
= [
Z
1
Q
(m)
] is an m×m permutat ion matrix. Q
(m)
is not unique but for any such
choice of Q
(m)
,
Z
T
1
Q
(m)
= 0, Q
T
(m)
Q

(m)
= I
m−r
, Q
(m)
Q
T
(m)
+ Z
1
Z
T
1
= I
m
(3.1)
(ii) for any m
×n matrix L,
Z
T
1
LL
∗
Z
1
≥

Z
T
1

LZ
2

Z
T
1
LZ
2

∗
≥ 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 of Z
1
that it may be considered as part of an m×m
permutation matrix Ω
= [
Z
1
Q
(m)
], where Q
(m)
is an m ×(m −r) matrix of zeros and ones.
For example, when k
= 2, then Q
(2)
is not unique (see, [8, page 49]). Using the properties
of a permutation matrix together with the definition of Ω

= [
Z
1
Q
(m)
], we have
I
m
= ΩΩ
T
=

Z
1
Q
(m)


Z
T
1
Q
T
(m)

=
Q
(m)
Q
T

(m)
+ Z
1
Z
T
1
,
I
m
=

I
r
0
0 I
m−r

=
Ω
T
Ω =

Z
T
1
Q
T
m



Z
1
Q
(m)

=

Z
T
1
Z
1
Z
T
1
Q
(m)
Q
T
(m)
Z
1
Q
T
(m)
Q
(m)

.
(3.3)

From these come the required results in (i), that is,
Z
T
1
Q
(m)
= 0, Q
T
(m)
Q
(m)
= I
m−r
, Q
(m)
Q
T
(m)
+ Z
1
Z
T
1
= I
m
. (3.4)
(ii) By (2.13)ofLemma 2.2,wehaveI
n
≥ Z
2

Z
T
2
≥ 0andso
Z
T
1
LL
∗
Z
1
≥ Z
T
1
LZ
2
Z
T
2
L
∗
Z
1
=

Z
T
1
LZ
2


Z
T
1
LZ
2

∗
≥ 0. (3.5)
We now generalize [8, Theorem 4] to the case of Khatri-Rao product involving a finite
number of matrices.

Z. A. Al Zhour and A. Kilicman 9
Theorem 3.2. Let A
i
= [A
(i)
gh
] ∈M
m,n
(1 ≤i ≤ k, k ≥2) be partitioned matrices with A
(i)
gh
as the (g,h)th block submatrix. Let Z
1
be an m
k
×r matrix of zeros and ones that satisfies
(2.12)andletQ
(n)

be an n
k
×(n
k
−s) matrix of zeros and ones that satisfies (3.1). Then
k

i=1
∗(A
i
A
∗
i
) =

k

i=1
∗(A
i
)

k

i=1
∗A
i

∗
+ Z

T
1

k

i=1
ΘA
i

Q
(n)
Q
T
(n)

k

i=1
ΘA
i

∗
Z
1
=

k

i=1
∗(A

i
)

k

i=1
∗A
i

∗
+

Z
T
1

k

i=1
ΘA
i

Q
(n)

Z
T
1

k


i=1
ΘA
i

Q
(n)

∗
,
(3.6)
and hence
k

i=1
∗

A
i
A
∗
i

≥

k

i=1
∗(A
i

)

k

i=1
∗A
i

∗
, k =2,3, (3.7)
Proof. From Lemma 2.1(i) and (iii), we have
k

i=1
Θ

A
i
A
∗
i

=

k

i=1
ΘA
i


k

i=1
ΘA
i

∗
. (3.8)
But by Theorem 3.1(i), there exist an n
k
×s matr ix Z
2
of zeros and ones that satisfies
(2.12)andann
k
×(n
k
−s)matrixQ
(n)
of zeros and ones that satisfies (3.1)suchthat
Z
2
Z
T
2
+ Q
(n)
Q
T
(n)

= I
n
k
and
k

i=1
Θ

A
i
A
∗
i

=

k

i=1
ΘA
i


Z
2
Z
T
2
+ Q

(n)
Q
T
(n)


k

i=1
ΘA
i

∗
=

k

i=1
ΘA
i


Z
2
Z
T
2


k


i=1
ΘA
i

∗
+

k

i=1
ΘA
i


Q
(n)
Q
T
(n)


k

i=1
ΘA
i

∗
.

(3.9)
Since A
i
(1 ≤i ≤ k, k ≥ 2) are rectangular partitioned matrices of order m ×n,thendue
to (2.11)ofLemma 2.2 there exist two real matrices Z
1
and Z
2
of zeros and ones of order
m
k
×r and n
k
×s, respectively, such that
k

i=1
∗A
i
= Z
T
1

k

i=1
ΘA
i

Z

2
, k = 2,3, (3.10)
But because A
i
A
∗
i
(1 ≤ i ≤ k, k ≥ 2) are square matrices of order m ×m,thendueto
(2.12)ofLemma 2.2 there exists a real matrix Z
1
of zeros and ones of order m
k
×r such
that
k

i=1
∗

A
i
A
∗
i

=
Z
T
1


k

i=1
Θ

A
i
A
∗
i


Z
1
, k = 2,3, (3.11)
10 Generalization inequalities for Khatri-Rao product
Due to (3.9), (3.10), and (3.11), we have
k

i=1
∗

A
i
A
∗
i

=
Z

T
1

k

i=1
Θ

A
i
A
∗
i


Z
1
= Z
T
1

k

i=1
ΘA
i

Z
2
Z

T
2

k

i=1
ΘA
i

∗
Z
1
+ Z
T
1

k

i=1
ΘA
i


Q
(n)
Q
T
(n)



k

i=1
ΘA
i

∗
Z
1
=

Z
T
1

k

i=1
ΘA
i

Z
2

Z
T
1

k


i=1
ΘA
i

Z
2

∗
+ Z
T
1

k

i=1
ΘA
i


Q
(n)
Q
T
(n)


k

i=1
ΘA

i

∗
Z
1
=

k

i=1
∗

A
i


k

i=1
∗A
i

∗
+ Z
T
1

k

i=1

ΘA
i

Q
(n)
Q
T
(n)

k

i=1
ΘA
i

∗
Z
1
=

k

i=1
∗

A
i


k


i=1
∗A
i

∗
+

Z
T
1

k

i=1
ΘA
i

Q
(n)

Z
T
1

k

i=1
ΘA
i


Q
(n)

∗
.
(3.12)

If we put k = 2inTheorem 3.2, we obtain the following corollary.
Corollary 3.3. Let A
i
= [A
(i)
gh
] ∈M
m,n
(1 ≤i ≤2) be part itioned matrices with A
(i)
gh
as the
(g,h)th block submatrix. Let Z
1
be an m
2
×r matrix of zeros and ones that satisfies (1.8)
and let Q
(n)
be an n
2
×(n

2
−s) matrix of zeros and ones that satisfies (3.1). Then
A
1
A
∗
1
∗A
2
A
∗
2
=

A
1
∗A
2

A
1
∗A
2

∗
+ Z
T
1

A

1
ΘA
2

Q
(n)
Q
T
(n)

A
1
ΘA
2

∗
Z
1
, (3.13)
and hence
A
1
A
∗
1
∗A
2
A
∗
2

≥

A
1
∗A
2

A
1
∗A
2

∗
. (3.14)
Corollary 3.4. Let A
i
= [A
(i)
gh
] ∈M
m,n
(1 ≤i ≤k, k ≥ 2) be partitioned matrices with A
(i)
gh
as the (g,h)th block submatrix. Let Z
1
be an m
k
×r matrix of zeros and ones that satisfies
(2.12)andletQ

(n)
be an n
k
×(n
k
−s) matrix of zeros and ones that satisfies (3.1). Then the
following statements are equivalent:
(i)
k

i=1
∗

A
i
A
∗
i

=

k

i=1
∗

A
i



k

i=1
∗A
i

∗
, k = 2,3, ; (3.15)
(ii)
Z
T
1

k

i=1
ΘA
i

Q
(n)
= 0, k =2,3, ; (3.16)
Z. A. Al Zhour and A. Kilicman 11
(iii)
k

i=1
∗

A

i
X
i

=

k

i=1
∗

A
i


k

i=1
∗

X
i


, for X
i
∈ M
n,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 to Z
T
1
(

k
i
=1
ΘA
i
)Q
(n)
= 0. To arrive from (ii) to (iii), notice
that (ii) may be rewritten as Z
T
1
(

k
i
=1
ΘA
i
)Q
(n)
Q
T
(n)
= 0. By Theorem 3.1(i), there exist an
n

k
×s matrix Z
2
of zeros and ones that satisfies (2.12)andann
k
×(n
k
−s)matrixQ
(n)
of
zeros and ones that satisfies (3.1)suchthatQ
(n)
Q
T
(n)
= I
n
k
−Z
2
Z
T
2
, this becomes
Z
T
1

k


i=1
ΘA
i

=
Z
T
1

k

i=1
ΘA
i

Z
2
Z
T
2
. (3.18)
By postmultiplying by (

k
i
=1
ΘX
i
)Z
1

for any of the n ×m matrices X
i
(1 ≤i ≤k), we have
Z
T
1

k

i=1
Θ

A
i
X
i


Z
1
= Z
T
1

k

i=1
ΘA
i


Z
2
Z
T
2

k

i=1
ΘX
i

Z
1
, (3.19)
which is (iii) by (2.11)and(2.12)ofLemma 2.2. To arrive from (iii) to (i), assume (iii)
holds for all n
×m matrices X
i
(1 ≤i ≤k). It must therefore be true for X
i
= A
∗
i
(1 ≤i ≤
k), which is condition (i). Hence (iii) implies (3.6) which is (i). 
If we put k = 2inCorollary 3.4, we obtain the following corollary.
Corollary 3.5. Let A
i
= [A

(i)
gh
] ∈M
m,n
(1 ≤i ≤2) be part itioned matrices with A
(i)
gh
as the
(g,h)th block submatrix. Let Z
1
be an m
2
×r matrix of zeros and ones that satisfies (1.8)and
let Q
(n)
be an n
2
×(n
2
−s) matrix of zeros and ones that satisfies (3.1). Then the following
statements are equivalent:
(i)
A
1
A
∗
1
∗A
2
A

∗
2
=

A
1
∗A
2

A
1
∗A
2

∗
; (3.20)
(ii)
Z
T
1

A
1
ΘA
2

Q
(n)
= 0; (3.21)
(iii)

A
1
X
1
∗A
2
X
2
=

A
1
∗A
2

X
1
∗X
2

, for X
1
,X
2
∈ M
n,m
. (3.22)
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
k

i=1
∗A
r
i
≥

k

i=1
∗A
i

r
; (3.23)
12 Generalization inequalities for Khatri-Rao product
(ii) if 0
≤ r ≤ 1, then
k

i=1
∗A
r
i
≤

k


i=1
∗A
i

r
. (3.24)
Proof. If we put s
= 1, replace r by 1/r and A
i
by A
r
i
in [3, Theorem 3.1(i)], we obtain (i).
But, if we put s
=−1, replace r by 1/ −r and A
i
by A
−r
i
in [3, Theorem 3.1(i)], we obtain
(ii).

Remark 3.7. It is easy to give another proof of Theorem 3.6 by replacing A by

k
i
=1
ΘA
i

in Lemma 2.3 and applying (2.12)ofLemma 2.2.
Theorem 3.8. Let A
i
> 0 be compatible partitioned matrices such that

k
i
=1
ΘA
i
> 0(1≤
i ≤k, k ≥2).LetW and w be the largest and smallest eigenvalues of

k
i
=1
ΘA
i
,respectively.
Then
(i) for every real p>1 and p<0,
k

i=1
∗A
p
i
≤ μ

k


i=1
∗A
i

p
, k =2,3, , (3.25)
where
μ
=
δ
p
−δ
(p −1)(δ −1)

p −1
p
δ
p
−1
δ
p
−δ

p
, δ =
W
w
. (3.26)
While for every 0 <p<1, the reverse inequality holds in (3.25);

(ii) for every real p>1 and p<0,
k

i=1
∗A
p
i
−

k

i=1
∗A
i

p
≤ γI, k = 2,3, , (3.27)
where
γ
=
Ww
p
−wW
p
W −w
+(p
−1)

1
p

W
p
−w
P
W −w

p/(p−1)
. (3.28)
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 the
sake of convenience. In (2.20)and(2.22)ofLemma 2.4,setk
= 1andreplaceU by Z
T
,
U
∗
by Z,andX by

k
i
=1
ΘA
i
,whereZ, is the selection matrix of zeros and ones that
satisfies (2.12). By using Lemma 2.1(iv), we establish Theorem 3.8.

From ( 3.25), we have the following special cases:
(i) for p
= 2, we have
k


i=1
∗A
2
i
≤

(W + w)
2
4wW

k

i=1
∗A
i

2
, k =2,3, ; (3.29)
Z. A. Al Zhour and A. Kilicman 13
(ii) for p
=−1, we have
k

i=1
∗A
−1
i
≤


(W + w)
2
4wW

k

i=1
∗A
i

−1
, k = 2,3, (3.30)
From ( 3.27), we have the following special cases:
(i) for p
= 2, we have
k

i=1
∗A
2
i
−

k

i=1
∗A
i

2

≤
1
4
(W
−w)
2
{I}, k = 2,3, ; (3.31)
(ii) for p
=−1, we have
k

i=1
∗A
−1
i
−

k

i=1
∗A
i

−1
≤

√
W −
√
w

wW

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 [
AB
B
∗
D
]with
a positive (semi) definite matrix A
∈ M
m
,

AB
B
∗
D

≥
0iff D ≥B
∗
A
+
B, (4.1)
for any positive semidefinite matrix D
∈ M
n
. It is also known that if matrix A is square

and nonsingular, then A
+
= A
−1
and [
AB
B
∗
D
] ≥0ifandonlyifD ≥B
∗
A
−1
B .
Let Z
1
and Z
2
be the real matrices of zeros and ones of order m ×r and n ×s,respec-
tively, that satisfy (2.11)inLemma 2.2. Now another way to use Lemma 2.2 to generate
inequalities involving the Khatri-Rao product is by using the following obvious inequal-
ity:
TT
∗
=

T
1
T
2



T
∗
1
T
∗
2

=

T
1
T
∗
1
T
1
T
∗
2
T
2
T
∗
1
T
2
T
∗

2

≥
0, (4.2)
where T
1
and T
2
are n ×l and m ×l matrices, respectively. Note that T
1
T
∗
1
and T
2
T
∗
2
are
positive semidefinite (positive definite) matrices for every (nonsingular) complex matr i-
ces T
1
and T
2
. This leads to

Z
T
2
0

0 Z
T
1

T
1
T
∗
1
T
1
T
∗
2
T
2
T
∗
1
T
2
T
∗
2

Z
2
0
0 Z
1


=

Z
T
2
T
1
T
∗
1
Z
2
Z
T
2
T
1
T
∗
2
Z
1
Z
T
1
T
2
T
∗

1
Z
2
Z
T
1
T
2
T
∗
2
Z
1

≥
0, (4.3)
if and only if
Z
T
1
T
2
T
∗
2
Z
1
≥

Z

T
1
T
2
T
∗
1
Z
2

Z
T
2
T
1
T
∗
1
Z
2

+

Z
T
2
T
1
T
∗

2
Z
1

. (4.4)
14 Generalization inequalities for Khatri-Rao product
Therefore (4.4) can be considered to be more general than (3.2). In order to prove this we
set T
1
= I and T
2
= L in (4.4), we have
Z
T
1
LL
∗
Z
1
≥

Z
T
1
LI
∗
Z
2

Z

T
2
II
∗
Z
2

+

Z
T
2
IL
∗
Z
1

=

Z
T
1
LZ
2

Z
T
2
Z
2


+

Z
T
2
L
∗
Z
1

(Z
T
2
Z
2
= I)
=

Z
T
1
LZ
2

Z
T
1
LZ
2


∗
.
(4.5)
Returning to (4.4)and(3.2), it can be easily seen that various other choices of the
matrices T
1
, T
2
,andL are possible which lead to quite different inequalities involvi ng
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.
Theorem 4.1. Let A
1
and A
2
be compatible partitioned matrices. Then
A
1
A
∗
1
∗A
2
A
∗
2

+ A
2
A
∗
2
∗A
1
A
∗
1
+ A
1
A
∗
2
∗A
2
A
∗
1
+ A
2
A
∗
1
∗A
1
A
∗
2

≥

A
1
∗A
2
+ A
2
∗A
1

A
1
∗A
2

∗
+

A
2
∗A
1

∗

.
(4.6)
Proof. Set T
1

= IΘI and T
2
= A
1
ΘA
2
+ A
2
ΘA
1
. Then calculations show that
T
2
T
∗
2
= A
1
A
∗
1
ΘA
2
A
∗
2
+ A
2
A
∗

2
ΘA
1
A
∗
1
+ A
1
A
∗
2
ΘA
2
A
∗
1
+ A
2
A
∗
1
ΘA
1
A
∗
2
,
T
2
T

∗
1
= A
1
ΘA
2
+ A
2
ΘA
1
, T
1
T
∗
2
=

A
1
ΘA
2

∗
+

A
2
ΘA
1


∗
, T
1
T
∗
1
= IΘI.
(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)
A
2
1
∗A
2
2
≥

A
1
∗A
2

2
; (4.8)
(ii)

A
2
∗A
−2
≥

A ∗A
−1

2
if A is nonsingular; (4.9)
(iii)
I
∗A
2
≥

I ∗A

2
. (4.10)
Proof. (i) Set A
∗
1
= A
1
and A
∗
2
= A

2
in (3.14)ofCorollary 3 .3,weget(4.8).
(ii) Set A
1
= A and A
2
= A
−1
in (4.8), we get (4.9).
(iii) Set A
1
= I and A
2
= A in (4.8), we get (4.10). 
Corollary 4.3. Let A
i
> 0(1≤i ≤2) be compatible partit ioned matrices. Then

A
2
1
∗A
2
2

1/2
≥ A
1
∗A
2

. (4.11)
Z. A. Al Zhour and A. Kilicman 15
Proof. It follows immediately by (4.8)andL
¨
owner-Heinz theorem.

Theorem 4.4. Let A
i
≥ 0(1≤ i ≤ k, k ≥ 2) be compatible partitioned matrices and let
A
0
i
= A
1/2
i
A
+1/2
i
= A
+1/2
i
A
1/2
i
(1 ≤i ≤k). Then
2

k

i=1

∗A
0
i

+

A
1
∗
k

i=2
∗A
+
i

+

A
+
1
∗
k

i=2
∗A
i

≥


A
1
∗
k

i=2
∗A
0
i
+ A
0
1
∗
k

i=2
∗A
i

k

i=1
∗A
i

+

A
1
∗

k

i=2
∗A
0
i
+ A
0
1
∗
k

i=2
∗A
i

.
(4.12)
Proof. Since A
i
≥ 0(1≤ i ≤ k, k ≥ 2), then A
∗
i
= A
i
.SetT
1
=

k

i
=1
ΘA
1/2
i
and T
2
=
A
1/2
1
Θ

k
i
=2
ΘA
+1/2
i
+ A
+1/2
1
Θ

k
i
=2
A
1/2
i

.SinceA
1/2
i
A
1/2
i
= A
i
, A
+1/2
i
A
+1/2
i
= A
+
i
,andA
0
i
=
A
1/2
i
A
+1/2
i
= A
+1/2
i

A
1/2
i
(1 ≤i ≤k), then calculations show that
T
2
T
∗
2
= 2

k

i=1
ΘA
0
i

+

A
1
Θ
k

i=2
ΘA
+
i


+

A
+
1
Θ
k

i=2
ΘA
i

, T
1
T
∗
1
=
k

i=1
ΘA
i
,
T
2
T
∗
1
=


A
1
Θ
k

i=2
ΘA
0
i
+ A
0
1
Θ
k

i=2
ΘA
i

, T
1
T
∗
2
=

A
1
Θ

k

i=2
ΘA
0
i
+ A
0
1
Θ
k

i=2
ΘA
i

.
(4.13)
Substituting these into (4.4) and using Lemma 2.2,weget(4.12).

If we put k = 2andreplaceA
i
by A
r
i
(1 ≤i ≤2) in Theorem 4.4,weobtainthefollowing
theorem.
Theorem 4.5. Let A
1
≥ 0, A

2
≥ 0 be compatible partitioned and let r be any nonze ro real
number such that A
0
1
= A
r/2
1
A
+r/2
1
= A
+r/2
1
A
r/2
1
and A
0
2
= A
r/2
2
A
+r/2
2
= A
+r/2
2
A

r/2
2
. Then
2A
0
1
∗A
0
2
+ A
r
1
∗A
+r
2
+ A
+r
1
∗A
r
2
≥

A
r
1
∗A
0
2
+ A

0
1
∗A
r
2

A
r
1
∗A
r
2

+

A
r
1
∗A
0
2
+ A
0
1
∗A
r
2

.
(4.14)

If A
1
> 0, A
2
> 0inTheorem 4.5, we obtain the follow ing theorem.
Theorem 4.6. Let A
1
> 0, A
2
> 0 be compatible partitioned and let I be a compatible parti-
tioned identity matrix. Then for any nonzero real number r,
2I + A
r
1
∗A
−r
2
+ A
−r
1
∗A
r
2
≥

A
r
1
∗I + I ∗A
r

2

A
r
1
∗A
r
2

−1

A
r
1
∗I + I ∗A
r
2

. (4.15)
If we put r
= 1andA
1
= A
2
in Theorem 4.6, we obtain the following theorem.
Theorem 4.7. Let A>0 be compat ible partitioned and let I be a compatible partitioned
identity matr ix. Then
2I + A
∗A
−1

+ A
−1
∗A ≥ (A ∗I + I ∗A)(A ∗A)
−1
(A ∗I +I ∗A). (4.16)
16 Generalization inequalities for Khatri-Rao product
In particular, if I is a nonpartitioned ident ity matrix, then
2I + A
∗A
−1
+ A
−1
∗A ≥ 4(I ∗A)(A ∗A)
−1
(I ∗A). (4.17)
Theorem 4.8. Let A
1
> 0 and A
2
> 0 be compatible partitioned matrices. Then for any
nonzero real number r
A
r
1
∗A
−r
2
+ A
−r
1

∗A
r
2
+2I ≥

A
r/2
1
∗A
−r/2
2
+ A
−r/2
1
∗A
r/2
2

2
. (4.18)
In particular, if A
1
= A
2
= A, Then
A
r
∗A
−r
+ A

−r
∗A
r
+2I ≥

A
r/2
∗A
−r/2
+ A
−r/2
∗A
r/2

2
. (4.19)
Proof. Since A
1
> 0andA
2
> 0, then A
∗
1
= A
1
and A
∗
2
= A
2

.SetL = A
r/2
1
ΘA
−r/2
2
+
A
−r/2
1
ΘA
r/2
2
.Compute
Z
T
1
LL
∗
Z
1
= Z
T
1
LLZ
1
= Z
T
1


A
r/2
1
ΘA
−r/2
2
+ A
−r/2
1
ΘA
r/2
2

A
r/2
1
ΘA
−r/2
2
+ A
−r/2
1
ΘA
r/2
2

Z
1
= Z
T

1

A
r
1
ΘA
−r
2

Z
1
+ Z
T
1
(IΘI)Z
1
+ Z
T
1
(IΘI)Z
1
+ Z
T
1

A
−r
1
ΘA
r

2

Z
1
= A
r
1
∗A
−r
2
+2I + A
−r
1
∗A
r
2
.
(4.20)
Similarly,

Z
T
1
LZ
2

Z
T
1
LZ

2

∗
=

Z
T
1
LZ
2

2
=

Z
T
1

A
r/2
1
ΘA
−r/2
2
+ A
−r/2
1
ΘA
r/2
2


Z
2

2
=

A
r/2
1
∗A
−r/2
2
+ A
−r/2
1
∗A
r/2
2

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) for r
= 1, we have
A
∗

1
A
−1
2
+ A
−1
1
∗A
2
+2I ≥

A
1/2
1
∗A
−1/2
2
+ A
−1/2
1
∗A
1/2
2

2
; (4.22)
(ii) for r
= 2, we have
A
2

1
∗A
−2
2
+ A
−2
1
∗A
2
2
+2I ≥

A
1
∗A
−1
2
+ A
−1
1
∗A
2

2
. (4.23)
Z. A. Al Zhour and A. Kilicman 17
From ( 4.19), we have the following special cases:
(i) for r
= 1, we have
A

∗A
−1
+ A
−1
∗A +2I ≥

A
1/2
∗A
−1/2
+ A
−1/2
∗A
1/2

2
; (4.24)
(ii) for r
= 2, we have
A
2
∗A
−2
+ A
−2
∗A
2
+2I ≥

A ∗A

−1
+ A
−1
∗A

2
. (4.25)
Theorem 4.9. Let A
1
≥ 0, A
2
≥ 0 be compatible partitioned and let I be a compatible p ar-
titioned identity matrix. Then
A
2
1
∞A
2
2
+2(A
1
∗A
2
) ≥

A
1
∞A
2


2
, (4.26)
where A
1
∞A
2
= A
1
∗I + I ∗A
2
is called the Khatri-Rao sum.
Proof. Set L
= A
1
∇A
2
= A
1
ΘI + IΘA
2
(Tracy-Singh sum). Since A
1
≥ 0andA
2
≥ 0, then
A
∗
1
= A
1

and A
∗
2
= A
2
. Calculations show that
Z
T
1
LL
∗
Z
1
= Z
T
1
LLZ
1
= Z
T
1

A
1
ΘI + IΘA
2

A
1
ΘI + I

2
ΘA
2

Z
1
= A
2
1
∗I + I ∗A
2
2
+2

A
1
∗A
2

=
A
2
1
∞A
2
2
+2

A
1

∗A
2

.
(4.27)
Similarly,

Z
T
1
LZ
2

Z
T
1
LZ
2

∗
=

Z
T
1

A
1
ΘI + IΘA
2


Z
2

Z
T
1

A
1
ΘI + IΘA
2

Z
2

∗
=

A
1
∗I + I ∗A
2

2
=

A
1
∞A

2

2
.
(4.28)
Substituting (4.27)and(4.28)into(3.2), we get (4.26).

Theorem 4.10. Let A
1
> 0 and A
2
> 0 be compatible partitioned matrices. Then for any
positive real number r,
r

A
2
1
∗A
2
2

+

A
1
A
2
∗A
2

A
1

+

A
2
A
1
∗A
1
A
2

+
1
r

A
2
2
∗A
2
1

≥
r

A
1

∗A
2

2
+

A
1
∗A
2

A
2
∗A
1

+

A
2
∗A
1

A
1
∗A
2

+
1

r

A
2
∗A
1

2
.
(4.29)
Proof. Set L
= ε
1
A
1
ΘA
2
+ ε
2
A
2
ΘA
1
,whereε
1
and ε
2
are both positive. Since A
1
> 0and

A
2
> 0, then A
∗
1
= A
1
and A
∗
2
= A
2
.Compute
Z
T
1
LL
∗
Z
1
= Z
T
1
LLZ
1
= Z
T
1

ε

1
A
1
ΘA
2
+ ε
2
A
2
ΘA
1

ε
1
A
1
ΘA
2
+ ε
2
A
2
ΘA
1

Z
1
= Z
T
1


ε
2
1

A
2
1
ΘA
2
2

+ ε
1
ε
2

A
1
A
2
ΘA
2
A
1

+ ε
1
ε
2


A
2
A
1
ΘA
1
A
2

+ ε
2
2

A
2
2
ΘA
2
1

Z
1
=

ε
2
1

A

2
1
∗A
2
2

+ ε
1
ε
2

A
1
A
2
∗A
2
A
1

+ ε
1
ε
2

A
2
A
1
∗A

1
A
2

+ ε
2
2

A
2
2
∗A
2
1

.
(4.30)
18 Generalization inequalities for Khatri-Rao product
Similarly,

Z
T
1
LZ
2

Z
T
1
LZ

2

∗
=

Z
T
1

ε
1
A
1
ΘA
2
+ ε
2
A
2
ΘA
1

Z
2

Z
T
1

ε

1
A
1
ΘA
2
+ ε
2
A
2
ΘA
1

Z
2

∗
=

Z
T
1

ε
1
A
1
ΘA
2
+ ε
2

A
2
ΘA
1

Z
2

Z
T
1

ε
1
A
1
ΘA
2
+ ε
2
A
2
ΘA
1

Z
2

=


ε
1
A
1
∗A
2
+ ε
2
A
2
∗A
1

2
= ε
2
1

A
1
∗A
2

2
+ ε
1
ε
2

A

1
∗A
2

A
2
∗A
1

+ ε
1
ε
2

A
2
∗A
1

A
1
∗A
2

+ ε
2
2

A
2

∗A
1

2
.
(4.31)
Substituting (4.30)and(4.31)into(3.2), we have

ε
2
1

A
2
1
∗A
2
2

+ ε
1
ε
2

A
1
A
2
∗A
2

A
1

+ ε
1
ε
2

A
2
A
1
∗A
1
A
2

+ ε
2
2

A
2
2
∗A
2
1

≥


ε
2
1

A
∗
1
A
2

2
+ ε
1
ε
2

A
1
∗A
2

A
2
∗A
1

+ ε
1
ε
2


A
2
∗A
1

A
1
∗A
2

+ ε
2
2

A
∗
2
A
1

2

.
(4.32)
Set r
= ε
1
/ε
2

,weget(4.29). 
Remark 4.11. Let A
i
(1 ≤ i ≤ k, k ≥ 2) be compatible partitioned matrices. Then (3.7)
can be proved by setting T
1
=

k
i
=1
ΘI and T
2
=

k
i
=1
ΘA
i
. Calculations show that
T
2
T
∗
2
=
k

i=1

ΘA
i
A
∗
i
, T
2
T
∗
1
=
k

i=1
ΘA
i
, T
1
T
∗
2
=

k

i=1
ΘA
i

∗

, T
1
T
∗
1
=
k

i=1
ΘI.
(4.33)
Substituting these into (4.4) and using (2.11), we get (3.7).
Remark 4.12. Let A
i
(1 ≤i ≤ 2) be compatible partitioned matrices. Then (3.14)canbe
proved by putting k
= 2inRemark 4.11.
Remark 4.13. All results obtained in Sections 3 and 4 are quite general. These results
lead to inequalities involving Hadamard product, as a special case, for nonpartitioned
matrices A
i
(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 commutativ ity of the Hadamard product to develop, for instance,
(3.7)ofTheorem 3.2. This result leads to the follow ing inequality involving Hadamard
product, as a special case:
k

i=1
◦


A
i
A
∗
i

≥

k

i=1
◦

A
i


k

i=1
◦A
i

∗
. (4.34)
It is possible to develop (4.34)inadifferent direction from (3.6). For example, Visick [8,
Theorem 11, page 54] proved that if A
1
, A

2
∈ M
m,n
and s ∈[−1,1], then
A
1
A
∗
1
◦A
2
A
∗
2
+ s

A
1
A
∗
2
◦A
2
A
∗
1

≥
(1 + s)


A
1
◦A
2

A
1
◦A
2

∗
. (4.35)
Z. A. Al Zhour and A. Kilicman 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 in Lemma 2.2, respectively, we obtain the following corollary.
Corollary 4.14. Let A
i
∈ M
m,n
(1 ≤i ≤k, k ≥2). Then
k

i=1
◦A
i
= P
T
km


k

i=1
⊗A
i

P
kn
, (4.36)
where P
km
= (
E
(m)
11
0
(m)
··· 0
(m)
E
(m)
22
0
(m)
··· 0
(m)
··· 0
(m)
··· 0

(m)
E
(m)
mm
)
T
is of order m
k
×m, 0
(m)
is an
m
×m matr ix with all entries equal to zero, and E
(m)
ij
is an m ×m matrix of zeros except for
aoneinthe(i, j)th position.
Theorem 4.15. Let A
i
∈ M
m,n
(1 ≤ i ≤ k, k ≥ 2). Then for any real scalars α
1
,α
2
, ,α
k
which are not all zero,

α

2
1
+ ···+ α
2
k


k

i=1
◦

A
i
A
∗
i


+

k−1

r=1
μ
r
k

w=1
◦


A
w
A
∗
(w+r)



≥

α
1
+ ···+ α
k

2

k

i=1
◦A
i

k

i=1
◦A
i


∗
,
(4.37)
where μ
r
=

k
w
=1
α
w
α
(w+r)

and w + r ≡(w + r)

mod k with 1 ≤ (w + r)

≤ k.
Proof. Let
L
= α
1

A
1
⊗A
2
⊗···⊗A

k

+ α
2

A
2
⊗···⊗A
k
⊗A
1

+ ···+ α
k

A
k
⊗A
1
⊗···⊗A
k−1

,
(4.38)
where A
i
∈ M
m,n
(1 ≤i ≤k, k ≥ 2) and α
1

,α
2
, ,α
k
are real scalars which are not all zero.
Taking indices “modk,” Lemma 2.1(i), (iii) (by setting
⊗ instead of Θ)give
LL
∗
=
k

i=1
α
i

A
i
⊗A
i+1
⊗···⊗A
i−1

k

i=1
α
i

A

∗
i
⊗A
∗
i+1
⊗···⊗A
∗
i−1

=
α
2
1

A
1
A
∗
1
⊗···⊗A
k
A
∗
k

+ ···+ α
2
k

A

k
A
∗
k
⊗AA
∗
1
⊗···⊗A
k
A
∗
k−1

+

i=j
α
i
α
j

A
i
A
∗
j
⊗A
j+1
A
∗

j+1
⊗···⊗A
j−1
A
∗
j−1

.
(4.39)
Now the application of (4.36) and the commutativity of the Hadamard product yield
P
T
km
LL
∗
P
km
=

α
2
1
+ ···+ α
2
k


k

i=1

◦

A
i
A
∗
i


+

k−1

r=1
μ
r
k

w=1
◦

A
w
A
∗
(w+r)



, (4.40)

where μ
r
=

k
w
α
w
α
(w+r)

and w + r ≡(w + r)

mod k with 1 ≤(w + r)

≤ k.
20 Generalization inequalities for Khatri-Rao product
Also by (4.36) and the commutativity of the Hadamard product, we obtain

P
T
km
LP
kn

=
P
T
km


α
1

A
1
⊗A
2
⊗···⊗A
k

+ α
2

A
2
⊗···⊗A
k
⊗A
1

+ ···+ α
k

A
k
⊗A
1
⊗···⊗A
k−1


P
kn
= α
1
P
T
km

A
1
⊗A
2
⊗···⊗A
k

P
kn
+ α
2
P
T
km
(A
2
⊗···⊗A
k
⊗A
1

P

kn
+ ···+ α
k
P
T
km

A
k
⊗A
1
⊗···⊗A
k−1

P
kn
= α
1

A
1
◦A
2
◦···◦A
k

+ α
2

A

2
◦···◦A
k
◦A
1

+ ···+ α
k

A
k
◦A
1
◦···◦A
k−1

=

α
1
+ ···+ α
k


k

i=1
◦A
i


,

P
T
km
LP
kn

∗
=

α
1
+ ···+ α
k


k

i=1
◦A
i

∗
.
(4.41)
Now

P
T

km
LP
kn

P
T
km
LP
kn

∗
=

α
1
+ ···+ α
k

2

k

i=1
◦A
i

k

i=1
◦A

i

∗
. (4.42)
Since P
T
km
LL
∗
P
km
≥ (P
T
km
LP
kn
)(P
T
km
LP
kn
)
∗
by (3.2)andfrom(4.40)and(4.42), we get
(4.37).

Now, we examine some special cases briefly.
In order to see that (4.37)reallyisanextensionin(4.34), it is sufficient to set α
1
= 1

and α
2
=···=α
k
= 0. Thus we recover the result of Visick in (4.35) which we mentioned
before the statement of Co r ollary 4.14.Letk
= 2, then μ
1
=

2
w
=1
α
w
α
(w+1)

with w +1≡
(w +1)

mod 2, that is, μ
1
= 2α
1
α
2
.ThenTheorem 4.15 asserts that

α

2
1
+ α
2
2

A
1
A
∗
1
◦A
2
A
∗
2

+2α
1
α
2

A
1
A
∗
2
◦A
2
A

∗
1

≥

α
1
+ α
2

2

A
1
◦A
2

A
1
◦A
2

∗
.
(4.43)
Simplification gives
A
1
A
∗

1
◦A
2
A
∗
2
+ s

A
1
A
∗
2
◦A
2
A
∗
1

≥
(1 + s)

A
1
◦A
2

A
1
◦A

2

∗
(4.44)
for any s
∈ [−1,1], just as we wanted. Finally, we present an attractive inequality using
three matrices. Let k
= 3, α
1
= 1, α
2
= α
3
=−1/2. Theorem 4.15 asserts that
A
1
A
∗
1
◦A
2
A
∗
2
◦A
3
A
∗
3
≥

1
2

A
1
A
∗
2
◦A
2
A
∗
3
◦A
3
A
∗
1
+ A
2
A
∗
1
◦A
3
A
∗
2
◦A
1

A
∗
3

. (4.45)
Z. A. Al Zhour and A. Kilicman 21
5. Acknowledgments
The authors would like to thank the referees for their valuable comments and sugges-
tions, including the simplified proof of Theorem 3.6 and some statements. The present
research has been partially supported by University Putra Malaysia (UPM) under the
Grant IRPA09-02-04-0259-EA001.
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ˇ
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Zeyad Abdel Aziz Al Zhour: Department of Mathematics and Institute for Mathematical Research,
University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
E-mail address:
Adem Kilicman: Department of Mathematics and Institute for Mathematical Research,
University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
E-mail address: