Journal of Inequalities in Pure and
Applied Mathematics
MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO
AND TRACY-SINGH SUMS
volume 7, issue 1, article 34,
2006.
ZEYAD AL ZHOUR AND ADEM KILICMAN
Department of Mathematics and Institute for Mathematical Research
University Putra Malaysia (UPM), Malaysia
43400, Serdang, Selangor, Malaysia
Received 05 August, 2004;
accepted 02 September, 2005.
Communicated by: R.P. Agarwal
EMail: zeyad1968@yahoo.com
EMail: akilic@fsas.upm.edu.my
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
148-04
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Abstract
The Khatri-Rao and Tracy-Singh products for partitioned matrices are viewed as
generalized Hadamard and generalized Kronecker products, respectively. We
define the Khatri-Rao and Tracy-Singh sums for partitioned matrices as generalized Hadamard and generalized Kronecker sums and derive some results
including matrix equalities and inequalities involving the two sums. Based on
the connection between the Khatri-Rao and Tracy-Singh products (sums) and
use mainly Liu’s, Mond and Pečarić’s methods to establish new inequalities involving the Khatri-Rao product (sum). The results lead to inequalities involving
Hadamard and Kronecker products (sums), as a special case.
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
2000 Mathematics Subject Classification: 15A45; 15A69.
Key words: Kronecker product (sum), Hadamard product (sum), Khatri-Rao product
(sum), Tracy-Singh product (sum), Positive (semi)definite matrix, Unitarily invariant norm, Spectral norm, P-norm, Moore-Penrose inverse.
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Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Basic Definitions on Matrix Products . . . . . . . . . . . . .
2.2
Basic Connections and Results on Matrix Products . .
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
On the Tracy-Singh Sum . . . . . . . . . . . . . . . . . . . . . . . .
3.2
On the Khatri-Rao Sum . . . . . . . . . . . . . . . . . . . . . . . . .
4
Special Results on Hadamard and Kronecker Sums . . . . . . . .
References
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
1.
Introduction
The Hadamard and Kronecker products are studied and applied widely in matrix
theory, statistics, econometrics and many other subjects. Partitioned matrices
are often encountered in statistical applications.
For partitioned matrices, The Khatri-Rao product viewed as a generalized
Hadamard product, is discussed and used in [7, 6, 14] and the Tracy-Singh product, as a generalized Kronecker product, is discussed and applied in [7, 5, 12].
Most results provided are equalities associated with the products. Rao, Kleffe
and Liu in [13, 8] presented several matrix inequalities involving the Khatri-Rao
product, which seem to be most existing results. In [7], Liu established the connection between Khatri-Rao and Tracy-Singh products based on two selection
matrices Z1 and Z2 . This connection play an important role to give inequalities involving the two products with statistical applications. In [10], Mond and
Pečarić presented matrix versions, with matrix weights. In [2, (2004)], Hiai and
Zhan proved the following inequalities:
(*)
kABk
kA + Bk
≤
,
kAk · kBk
kAk + kBk
kA ◦ Bk
kA + Bk
≤
kAk · kBk
kAk + kBk
for any invariant norm with kdiag(1, 0, . . . , 0)k ≥ 1 and A, B are nonzero
positive definite matrices.
In the present paper, we make a further study of the Khatri-Rao and TracySingh products. We define the Khatri-Rao and Tracy-Singh sums for partitioned
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
matrices and use mainly Liu’s, Mond and Pečarić’s methods to obtain new inequalities involving these products (sums).We collect several known inequalities which are derived as a special cases of some results obtained. We generalize
the inequalities in Eq (*) involving the Hadamard product (sum) and the Kronecker product (sum).
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
2.
2.1.
Basic Definitions and Results
Basic Definitions on Matrix Products
We introduce the definitions of five known matrix products for non-partitioned
and partitioned matrices. These matrix products are defined as follows:
Definition 2.1. Consider matrices A = (aij ) and C = (cij ) of order m × n and
B = (bkl ) of order p × q. The Kronecker and Hadamard products are defined
as follows:
1. Kronecker product:
A ⊗ B = (aij B)ij ,
(2.1)
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
where aij B is the ijth submatrix of order p×q and A⊗B of order mp×nq.
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2. Hadamard product:
A ◦ C = (aij cij )ij ,
(2.2)
th
where aij cij is the ij scalar element and A ◦ C is of order m × n.
Definition 2.2. Consider matrices A = (aij ) and B = (bkl ) of order m × m
and n × n respectively. The Kronecker sum is defined as follows:
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(2.3)
A ⊕ B = A ⊗ In + Im ⊗ B,
where In and Im are identity matrices of order n × n and m × m respectively,
and A ⊕ B of order mn × mn.
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Definition 2.3. Consider matrices A and C of order m×n, and B of order p×q.
Let A = (Aij ) be partitioned with Aij of order mi ×nj as the ijth submatrix,C =
(Cij ) be partitioned with Cij of order mi × nj as the ijth submatrix, and B =
(B
p k × ql P
as the klth submatrix, where,m =
Prkl ) be partitioned
Ps with Bkl ofPorder
t
h
i=1 mi , n =
j=1 nj , p =
k=1 pk , q =
l=1 ql are partitions of positive
integers m, n, p, and q. The Tracy-Singh and Khatri-Rao products are defined
as follows:
1. Tracy-Singh product:
(2.4)
AΠB = (Aij ΠB)ij = (Aij ⊗ Bkl )kl
ij
,
where Aij is the ijth submatrix of order mi × nj , Bkl is the klth submatrix
of order pk × ql , Aij ΠB is the ijth submatrix of order mi p × nj q, Aij ⊗ Bkl
is the klth submatrix of order mi pk × nj ql and AΠB of order mp × nq.
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
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Note that
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(i) For a non partitioned matrix A, their AΠB is A ⊗ B, i.e., for A =
(aij ), where aij is scalar, we have,
AΠB = (aij ΠB)ij
= (aij ⊗ Bkl )kl ij
= (aij Bkl )kl ij = (aij B)ij = A ⊗ B.
(ii) For column wise partitioned A and B, their AΠB is A ⊗ B.
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
2. Khatri-Rao product:
(2.5)
A ∗ B = (Aij ⊗ Bij )ij ,
where Aij is the ijth submatrix of order mi × nj , Bij is the ijth submatrix
of order pi × qj , Aij ⊗ B
ijth submatrix of order mi pi × nj qj and
ij is the
P
P
A ∗ B of order M × N M = ri=1 mi pi , N = sj=1 nj qj .
Note that
(i) For a non partitioned matrix A, their A ∗ B is A ⊗ B, i.e., for A =
(aij ),where aij is scalar, we have,
A ∗ B = (aij ⊗ Bij )ij = (aij B)ij = A ⊗ B.
(ii) For non partitioned matrices A and B, their A ∗ B is A ◦ B, i.e., for
A = (aij ) and B = (bij ), where aij and bij are scalars, we have,
A ∗ B = (aij ⊗ bij )ij = (aij bij )ij = A ◦ B.
2.2.
Basic Connections and Results on Matrix Products
We introduce the connection between the Katri-Rao and Tracy-Singh products
and the connection between the Kronecker and Hadamard products, as a special case, which are important in creating inequalities involving these products.
We write A ≥ B in the Löwner ordering sense that A − B ≥ 0 is positive
semi-definite, for symmetric matrices A and B of the same order and A+ and
A∗ indicate the Moore-Penrose inverse and the conjugate of the matrix A, respectively.
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Lemma 2.1. Let A = (aij ) and B = (bij ) be two scalar matrices of order
m × n. Then (see [15])
(2.6)
A ◦ B = K10 (A ⊗ B)K2
where K1 and K2 are two selection matrices of order n2 × n and m2 × m,
respectively, such that K10 K1 = Im and K20 K2 = In .
In particular, for m = n, we have K1 = K2 = K and
(2.7)
A ◦ B = K 0 (A ⊗ B)K
Lemma 2.2. Let A and B be compatibly partitioned. Then (see [8, p. 177-178]
and [7, p. 272])
(2.8)
Zeyad Al Zhour and
Adem Kilicman
A ∗ B = Z10 (AΠB) Z2 ,
where Z1 and Z2 are two selection matrices of zeros and ones such that Z10 Z1 =
I1 and Z20 Z2 = I2 , where I1 and I2 are identity matrices.
In particular, when A and B are square compatibly partitioned matrices, then
we have Z1 = Z2 = Z such that Z 0 Z = I and
(2.9)
Matrix Equalities and
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Khatri-Rao and Tracy-Singh
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A ∗ B = Z 0 (AΠB) Z.
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Note that, for non-partitioned matrices A, B, Z1 and Z2 , Lemma 2.2 leads to
Lemma 2.1, as a special case.
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Lemma 2.3. Let A, B, C, D and F be compatibly partitioned matrices. Then
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(AΠB)(CΠD) = (AC)Π(BD)
(AΠB)+ = A+ ΠB +
(A + C)Π(B + D) = AΠB + AΠD + CΠB + CΠD
(AΠB)∗ = A∗ ΠB ∗
AΠB 6= BΠA
in general
A ∗ B 6= B ∗ A
in general
B ∗ F = F ∗ B where F = (fij ) and fij is a scalar
(A ∗ B)∗ = A∗ ∗ B ∗
(A + C) ∗ (B + D) = A ∗ B + A ∗ D + C ∗ B + C ∗ D
(A ∗ B)Π(C ∗ D) = (AΠC) ∗ (BΠD)
Matrix Equalities and
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Proof. Straightforward.
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Lemma 2.4. Let A and B be compatibly partitioned matrices. Then
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(2.20)
(AΠB)r = Ar ΠB r ,
for any positive integer r.
Proof. The proof is by induction on r and using Eq. (2.10).
Theorem 2.5. Let A ≥ 0 and B ≥ 0 be compatibly partitioned matrices. Then
(2.21)
(AΠB)α = Aα ΠB α
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for any positive real α.
J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Proof. By using Eq (2.20), we have AΠB = (A1/n ΠB 1/n )n , for any positive
integer n. So it follows that (AΠB)1/n = A1/n ΠB 1/n . Now (AΠB)m/n =
Am/n ΠB m/n , for any positive integers n, m. The Eq (2.21) now follows by a
continuity argument.
Corollary 2.6. Let A and B be compatibly partitioned matrices. Then
|AΠB| = |A| Π |B| ,
(2.22)
where
|A| = (A∗ A)1/2
Proof. Applying Eq (2.10) and Eq (2.21), we get the result.
Theorem 2.7. Let A = (Aij ) and B = (BklP
) be partitionedPmatrices of order
m × m, and n × n respectively, where m = ri=1 mi , n = tk=1 nk .Then
Matrix Equalities and
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Khatri-Rao and Tracy-Singh
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Zeyad Al Zhour and
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(2.23)
(a)
tr(AΠB) = tr(A) · tr(B)
(2.24)
(b)
kAΠBkp = kAkp kBkp , where kAkp = [tr |A|p ]
1/p
,
for all 1 ≤ p < ∞.
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Proof. (a) Straightforward.
(b) Applying Eq (2.22) and Eq (2.23), we get the result.
Theorem 2.8. Let A, B and I be compatibly partitioned matrices. Then
(2.25)
(AΠI)(IΠB) = (IΠB)(AΠI) = AΠB.
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If f (A) is an analytic function on a region containing the eigenvalues of A, then
(2.26)
f (IΠA) = IΠf (A) and
f (AΠI) = f (A)ΠI
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Proof. The proof of Equation (2.25) is straightforward on applying Eq (2.10).
Equation (2.26) can be proved as follows:
P
k
Since f (A)is an analytic function, then f (A) = ∞
k=0 αk A . Applying Eq
(2.10) we get:
∞
∞
∞
X
X
X
k
k
f (IΠA) =
αk (IΠA) =
αk (IΠA ) = IΠ
αk Ak = IΠf (A).
k=0
k=0
k=0
Corollary 2.9. Let A, B and I be compatibly partitioned matrices. Then
(2.27)
eAΠI = eA ΠI
and
eIΠA = IΠeA .
Lemma 2.10. Let H ≥ 0 be a n × n matrix with nonzero eigenvalues λ1 ≥
· · · ≥ λk (k ≤ n) and X be a m × m matrix such that X = H 0 X, where
H 0 = HH + . Then (see [6, Section 2.3])
(λ1 + λk )2 0
0
(2.28)
(X 0 HX)+ ≤ X + H + X + ≤
(X HX)+ .
(4λ1 λk )
Theorem 2.11. Let A ≥ 0 and B ≥ 0 be compatibly partitioned matrices such
that A0 = AA+ and B 0 = BB + . Then (see [8, Section 3])
(2.29) (A∗B 0 +A0 ∗B)(A∗B)+ (A∗B 0 +A0 ∗B) ≤ A∗B + +A+ ∗B+2A0 ∗B 0
Theorem 2.12. Let A > 0 and B > 0 be n × n compatibly partitioned matrices
with eigenvalues contained in the interval between m and M (M ≥ m). Let I
be a compatible identity matrix. Then (see [8, Section 3]).
(2.30)
m2 + M 2
A ∗ B −1 + A−1 ∗ B ≤
I
mM
and
A ∗ A−1
m2 + M 2
≤
I
2mM
Matrix Equalities and
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
3.
Main Results
3.1.
On the Tracy-Singh Sum
Definition 3.1. Consider matrices A and B of order m × m and n × n respectively. Let A = (Aij ) be partitioned with Aij of order mi × mi as the ijth
submatrix, and let BP
= (Bij ) be partitioned
with Bij of order nk × nk as the
Pt
r
th
ij submatrix m = i=1 mi , n = k=1 nk .
The Tracy-Singh sum is defined as follows:
(3.1)
A∇B = AΠIn + Im ΠB,
where In = In1 +n2 +···+nt = blockdiag(In1 , In2 , . . . , Int ) is an n × n identity
matrix, Im = Im1 +m2 +···+mr = blockdiag(Im1 , Im2 , . . . , Imr ) is an m × m identity matrix, Ink is an nk × nk identity matrix (k = 1, . . . , t), Imi is an mi × mi
identity matrix (i = 1, . . . , r) and A∇B is of order mn × mn.
Note that for non-partitioned matrices A and B, their A∇B is A ⊕ B.
Theorem 3.1. Let A ≥ 0, B ≥ 0, C ≥ 0 and D ≥ 0 be compatibly partitioned
matrices. Then
(3.2)
(A∇B)(C∇D) ≥ AC∇BD.
Matrix Equalities and
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Proof. Applying Eq (3.1) and Eq (2.10), we have
(A∇B)(C∇D)
= (AΠI + IΠB)(CΠI + IΠD)
= (AΠI)(CΠI) + (AΠI)(IΠD) + (IΠB)(CΠI) + (IΠB)(IΠD)
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
= ACΠI + AΠD + CΠB + IΠBD
= AC∇BD + AΠD + CΠB ≥ AC∇BD.
In special cases of Eq (3.2), if C = A∗ , D = B ∗ , we have
(3.3)
(A∇B)(A∇B)∗ ≥ AA∗ ∇BB ∗
and if C = A, D = B, we have
(A∇B)2 ≥ A2 ∇B 2 .
(3.4)
More generally, it is easy by induction on w we can show that ifA ≥ 0 and
B ≥ 0 are compatibly partitioned matrices. Then
w−1 X
w
w
w
w
(3.5)
(A∇B) = A ∇B +
(Aw−k ΠB k );
k
k=1
Matrix Equalities and
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(A∇B)w ≥ Aw ∇B w
(3.6)
for any positive integer w.
Theorem 3.2. Let APand B be partitioned
matrices
of order m × m and n × n,
P
respectively, m = ri=1 mi , n = tk=1 nk . Then
(3.7)
tr(A∇B) = n · tr(A) + m · tr(B),
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(3.8)
kA∇Bkp ≤
√
p
n kAkp +
√
p
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m kBkp ,
J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
where kAkp = [tr |A|p ]1/p , 1 ≤ p < ∞, and
(3.9)
eA∇B = eA ΠeB .
Proof. For the first part, on applying Eq (2.23), we obtain
tr(A∇B) = tr [(AΠIn ) + (Im ΠB)]
= tr(AΠIn ) + tr(Im ΠB)
= tr(A) tr(In ) + tr(Im ) tr(B)
= n · tr(A) + m · tr(B).
To prove (3.8), we apply Eq (2.24), to get
Matrix Equalities and
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kA∇Bkp = k(AΠIn ) + (Im ΠB)kp
≤ kAΠIn kp + kIm ΠBkp
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= kAkp kIn kp + kIm kp kBkp
√
√
= p n kAkp + p m kBkp .
Contents
For the last part, applying Eq (2.25), Eq (2.27) and Eq (2.10), we have
eA∇B = e(AΠIn )+(Im ΠB)
= e(AΠIn ) e(Im ΠB)
= (eA ΠIn )(Im ΠeB ) = eA ΠeB .
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Theorem 3.3. Let A and Bbe non
matrices of order m×m
Pr singular partitioned
Pt
and n × n respectively, (m = i=1 mi , n = k=1 nk ).Then
(3.10)
(i)
(3.11) (ii)
(3.12) (iii)
(A∇B)−1 = (A−1 ∇B −1 )−1 (A−1 ΠB −1 )
(A∇B)−1 = (A−1 ΠIn )(A−1 ∇B −1 )−1 (Im ΠB −1 )
(A∇B)−1 = (Im ΠB −1 )(A−1 ∇B −1 )−1 (A−1 ΠIn )
Proof. (i) Applying Eq (2.10), we have
(A∇B)−1 = [Im ΠB + AΠIn ]−1
= [(Im ΠB)(Im ΠIn ) + (Im ΠB)(AΠB −1 )]−1
= [(Im ΠB)(Im ΠIn + AΠB −1 )]−1
= [(Im ΠIn + AΠB −1 )]−1 [Im ΠB]−1
= [(AΠIn )(A−1 ΠIn ) + (AΠIn )(Im ΠB −1 )]−1 [Im ΠB −1 ]
= [(AΠIn ){A−1 ΠIn + Im ΠB −1 }]−1 [Im ΠB −1 ]
= [(AΠIn )(A−1 ∇B −1 )]−1 [Im ΠB −1 ]
= (A−1 ∇B −1 )−1 (A−1 ΠIn )(Im ΠB −1 )
= (A−1 ∇B −1 )−1 (A−1 ΠB −1 ).
Similarly, we obtain (ii) and (iii).
Matrix Equalities and
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Theorem 3.4. Let A ≥ 0 and I be compatibly partitioned matrices such that
A+ ΠI = IΠA+ . Then
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A∇A+ ≥ 2AA+ ΠI.
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(3.13)
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Proof. We know that A∇I = AΠI + IΠI > AΠI. Denote H = M ΠI ≥ 0.
By virtue of H + H + ≥ 2HH + and Eq (2.10), we have
AΠI + (AΠI)+ ≥ 2(AΠI)(AΠI)+ = 2AA+ ΠI
Since, A+ ΠI = IΠA+ , we get the result.
3.2.
On the Khatri-Rao Sum
Definition 3.2. Let A, B, In and Im be partitioned as in Definition 3.1. Then
the Khatri-Rao sum is defined as follows:
(3.14)
A∞B = A ∗ In + Im ∗ B
Note that, for non-partitioned matrices A and B, their A∞B is A ⊕ B, and
for non-partitioned matrices A, B, In and Im , their A∞B is A • B (Hadamard
sum, see Definition 4.1, Eq(4.1), Section 4).
Theorem 3.5. Let A and B be compatibly partitioned matrices. Then
(3.15)
A∞B = Z 0 (A∇B)Z,
where Z is a selection matrix as in Lemma 2.2.
Proof. Applying Eq (2.9), we have A ∗ I = Z 0 (AΠI)Z, I ∗ B = Z 0 (IΠB)Z
and
A∞B = A ∗ I + I ∗ B = Z 0 (AΠI) Z + Z 0 (IΠB) Z = Z 0 (A∇B)Z.
Matrix Equalities and
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Corollary 3.6. Let A ≥ 0 and I be compatibly partitioned matrices such that
A+ ΠI = IΠA+ . Then
A∞A+ ≥ 2AA+ ∗ I
(3.16)
Proof. Applying Eq(3.13) and Eq (3.15), we get the result.
Corollary 3.7. Let A > 0 be compatibly partitioned with eigenvalues contained
in the interval between m and M (M ≥ m). Let I be a compatible identity
matrix such that A−1 ∞I = I∞A−1 . Then
A∞A−1 ≤
(3.17)
2
2
m +M
I.
mM
Proof. Applying Eq (2.30) and taking B = I, we get the result.
Zeyad Al Zhour and
Adem Kilicman
Corollary 3.8. Let A ≥ 0 and I be compatibly partitioned, where A0 = AA+
such that A0 ∗ I = I ∗ A0 . Then
(3.18)
0
+
0
+
0
(A∞A )(A ∗ I) (A∞A ) ≤ A ∗ I + A ∗ I + 2A ∗ I
and if A+ ∗ I = I ∗ A+ , we have
(3.19)
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
(A∞A0 )(A ∗ I)+ (A∞A0 ) ≤ A∞A+ + 2A0 ∗ I.
Proof. Applying Eq (2.29) and taking B = I, we get the results.
Mond and Pečarić (see [10]) proved the following result:
If Xj (j = 1, 2, . . . , k) are positive definite Hermitian matrices of order
n × n with eigenvalues in the interval [m, M ] and Uj (j = 1, 2, . . ., k) are r × n
P
matrices such that kj=1 Uj Uj∗ = I. Then
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
(a) For p < 0 or p > 1, we have
(3.20)
k
X
Uj Xjp Uj∗ ≤ λ
j=1
k
X
!p
Uj Xj Uj∗
j=1
where,
(3.21)
γp − γ
λ=
(p − 1)(γ − 1)
p(γ − γ p )
(1 − p)(γ p − 1)
−p
,
γ=
M
.
m
While, for 0 < p < 1, we have the reverse inequality in Eq (3.20).
(b) For p < 0 or p > 1, we have
(3.22)
k
X
!
Uj Xjp Uj∗
−
j=1
k
X
Zeyad Al Zhour and
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!p
Uj Xj Uj∗
≤ αI,
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j=1
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where,
(3.23)
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
α = mp −
M p − mp
p(M − m)
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J
p
p−1
M p − mp
+
(M − m)
"
M p − mp
p(M − m)
1
p−1
#
−m .
While, for 0 < p < 1, we have the reverse inequality in Eq (3.22).
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We have an application to the Khatri-Rao product and Khatri-Rao sum.
J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Theorem 3.9. Let A and B be positive definite Hermitian compatibly partitioned matrices and let m and M be, respectively, the smallest and the largest
eigenvalues of AΠB. Then
(a) For p a nonzero integer, we have
(3.24)
Ap ∗ B p ≤ λ(A ∗ B)p
where, λ is given by Eq (3.21).
While, for 0 < p < 1, we have the reverse inequality in Eq (3.24).
(b) For p a nonzero integer, we have
(3.25)
Zeyad Al Zhour and
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(Ap ∗ B p ) − (A ∗ B)p ≤ αI,
where α is given by Eq (3.23).
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While, for 0 < p < 1, we have the reverse inequality in Eq (3.25).
∗
Proof. In Eq (3.20) and Eq (3.22), take k = 1 and instead of U , use Z, the
selection matrix which satisfy the following property:
A ∗ B = Z 0 (AΠB)Z,
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Z 0 Z = I.
Making use of the fact in Eq (2.21) that for any real n (positive or negative), we
have
(AΠB)n = An ΠB n ,
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
then, with Z 0 , AΠB, Z substituted for U , X, U ∗ , we have from Eq (3.20)
Ap ∗ B p = Z 0 (Ap ∗ B p )Z
= Z 0 (A ∗ B)p Z
p
≤ λ {Z 0 (AΠB)Z} = λ(A ∗ B)p ,
where, λ is given by Eq (3.21)
Similarly, from Eq (3.22), we obtain for
(Ap ∗ B p ) − (A ∗ B)p ≤ αI
where, α is given by Eq (3.23).
Special cases include from Eq (3.24):
(2.1) For p = 2, we have
(3.26)
A2 ∗ B 2 ≤
(M + m)2
{A ∗ B}2
4M m
(2.2) For p = −1, we have
(3.27)
A−1 ∗ B −1 ≤
(M + m)2
{A ∗ B}−1
4M m
Similarly, special cases include from Eq (3.25):
(2.1) For p = 2, we have
(3.28)
1
(A2 ∗ B 2 ) − (A ∗ B)2 ≤ (M − m)2 I
4
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
(2.2) For p = −1, we have
√
(3.29)
(A
−1
−1
−1
∗ B ) − (A ∗ B)
≤
√
M− m
{I} ,
Mm
where results in Eq (3.26), Eq (3.27), and Eq (3.28) are given in [7].
Theorem 3.10. Let A and B be positive definite Hermitian compatibly partitioned matrices. Let m1 and M1 be, respectively, the smallest and the largest
eigenvalues of AΠI and m2 and M2 , respectively, the smallest and the largest
eigenvalues of IΠB. Then
(a) For p a nonzero integer, we have
Zeyad Al Zhour and
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Ap ∞B p ≤ max {λ1 , λ2 } (A∞B)p
(3.30)
where,
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(3.31) λ1 =
(3.32)
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
(γ1p
− γ1 )
[(p − 1)(γ1 − 1)]
(γ2p − γ2 )
λ2 =
[(p − 1)(γ2 − 1)]
−p
p(γ1 − γ1p )
[(1 − p)(γ1p − 1)]
p(γ2 − γ2p )
[(1 − p)(γ2p − 1)]
,
γ1 =
−p
,
γ2 =
While, for 0 < p < 1, we have the reverse inequality in Eq (3.30).
(b) For p a nonzero integer, we have
M1
,
m1
M2
.
m2
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(3.33)
(Ap ∞B p ) − (A∞B)p ≤ max {α1 , α2 } I
J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
where,
(3.34)
α1 =
mp1
−
M1p − mp1
p(M1 − m1 )
p
p−1
M p − mp1
+ 1
M1 − m 1
(3.35)
α2 =
mp2
−
M2p − mp2
p(M2 − m2 )
(
M1p − mp1
p(M1 − m1 )
1
p−1
)
− m1
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
p
p−1
M2p − mp2
+
M2 − m 2
(
M2p − mp2
p(M2 − m2 )
1
p−1
)
− m2
While, for 0 < p < 1, we have the reverse inequality in Eq (3.33).
Proof. Applying Eq (3.24), we have
Ap ∗ I = Ap ∗ I p ≤ λ1 (A ∗ I)p
I ∗ B p = I p ∗ B p ≤ λ2 (I ∗ B)p
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Ap ∞B p = Ap ∗ I + I ∗ B p
≤ λ1 (A ∗ I)p + λ2 (I ∗ B)p
≤ max {λ1 , λ2 } [A ∗ I + I ∗ B]p = max {λ1 , λ2 } (A∞B)p
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
where, λ1 and λ2 are given in Eq (3.31) and Eq (3.32).
Similarly, from Eq (3.25), we obtain for
(Ap ∞B p ) − (A∞B)p ≤ max {α1 , α2 } I
where, α1 and α2 are given in Eq (3.34) and Eq (3.35).
Special cases include from Eq (3.30):
(2.1) For p = 2, we have
(M1 + m1 )2 (M2 + m2 )2
2
2
(3.36)
A ∞B ≤ max
,
{A∞B}2 .
4M1 m1
4M2 m2
Zeyad Al Zhour and
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(2.2) For p = −1, we have
(3.37)
−1
A ∞B
−1
≤ max
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
(M1 + m1 )2 (M2 + m2 )2
,
4M1 m1
4M2 m2
{A∞B}−1 .
Similarly, special cases include from Eq (3.33):
(2.1) For p = 2, we have
1
2
2
2
2 1
2
(3.38) (A ∞B ) − (A∞B) ≤ max
(M1 − m1 ) , (M2 − m2 ) I.
4
4
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(2.2) For p = −1, we have
(3.39) (A−1 ∞B −1 ) − (A∞B)−1 ≤ max
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√
√
√
√ M1 − m 1 M 2 − m 2
,
I.
4M1 m1
4M2 m2
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Theorem 3.11. Let A and B be positive definite Hermitian compatibly partitioned matrices. Let m and M be, respectively, the smallest and the largest
eigenvalues of A∇B. Then
(a) For p a nonzero integer, we have
(3.40)
Ap ∞B p ≤ λ(A∞B)p ,
where λ is given by Eq (3.21).
While, for 0 < p < 1, we have the reverse inequality in Eq (3.40).
(b) For p a nonzero integer, we have
(3.41)
(Ap ∞B p ) − (A∞B)p ≤ αI
where, α is given by Eq (3.23).
While, for 0 < p < 1, we have the reverse inequality in Eq (3.41).
Proof. In Eq (3.20) and Eq (3.22), take k = 1 and instead of U ∗ , use Z, the
selection matrix which satisfy the following property:
A∞B = Z 0 (A∇B)Z,
Z 0Z = I
Then, with Z 0 , A∇B, Z substituted for U , X, U ∗ , we have from Eq (3.20)
Ap ∞B p = Z 0 (Ap ∇B p )Z
= Z 0 (Ap ΠI + IΠB p )Z
≤ Z 0 {A∇B}p Z
p
≤ λ {Z 0 (A∇B)Z} = λ(A∞B)p
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
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where, λ is given by Eq (3.21).
J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Similarly, from Eq (3.22), we obtain Eq (3.41)
Special cases include from Eq (3.40):
(2.1) For p = 2, we have
A2 ∞B 2 ≤
(3.42)
(M + m)2
{A∞B}2
4M m
(2.2) For p = −1, we have
(3.43)
−1
A ∞B
−1
(M + m)2
≤
{A∞B}−1
4M m
Similarly, special cases include from Eq (3.41):
(2.1) For p = 2, we have
(3.44)
Zeyad Al Zhour and
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1
(A2 ∞B 2 ) − (A∞B)2 ≤ (M − m)2 I
4
√
(3.45)
−1
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(2.2) For p = −1, we have
−1
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
−1
(A ∞B ) − (A∞B)
≤
√
M− m
{I}
Mm
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
4.
Special Results on Hadamard and Kronecker
Sums
The results obtained in Section 3 are quite general. Now, we consider some
inequalities in a special case which involves non-partitioned matrices A, B and
I with the Hadamard product (sum) replacing the Khatri-Rao product (sum)
and the Kronecker product (sum) replacing the Tracy-Singh product (sum). As
these inequalities can be viewed as a corollary (some of) the proofs are straightforward and alternative to those for the existing inequalities.
Definition 4.1. Let A and B be square matrices of order n × n.The Hadamard
sum is defined as follows:
(4.1)
A • B = A ◦ In + In ◦ B = A ◦ In + B ◦ In = (A + B) ◦ In .
Corollary 4.1. Let A > 0. Then
Corollary 4.2. Let A > 0 be a matrix of order n×n with eigenvalues contained
in the interval between m and M (M ≥ m). Then
(4.3)
A•A
−1
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A • A−1 ≥ 2I.
(4.2)
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
(m2 + M 2 )
{I} .
≤
mM
Corollary 4.3. Let A and B be n × n positive definite Hermitian matrices and
let m and M be, respectively, the smallest and the largest eigenvalues of A ⊗ B.
Then
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
(a) For p a nonzero integer, we have
Ap ◦ B p ≤ λ(A ◦ B)p
(4.4)
where, λ is given by Eq (3.21).
While, for 0 < p < 1, we have the reverse inequality in Eq (4.4).
(b) For p is a nonzero integer, we have
(Ap ◦ B p ) − (A ◦ B)p ≤ αI
(4.5)
where, α is given by Eq (3.23).
While, for 0 < p < 1, we have the reverse inequality in Eq (4.5).
Special cases include from Eq (4.4):
(2.1) For p = 2, we have
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
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(4.6)
(M + m)2
A2 ◦ B 2 ≤
{A ◦ B}2
4M m
(2.2) For p = −1, we have
(4.7)
A−1 ◦ B −1 ≤
(M + m)2
{A ◦ B}−1 .
4M m
Similarly, special cases include from Eq (4.5):
(2.1) For p = 2, we have
(4.8)
1
(A2 ◦ B 2 ) − (A ◦ B)2 ≤ (M − m)2 I
4
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
(2.2) For p = −1, we have
√
(4.9)
(A
−1
−1
−1
◦ B ) − (A ◦ B)
≤
√
M− m
{I} ,
Mm
where results in Eq (4.6), Eq (4.7), and Eq (4.8) are given in [11].
We note that the eigenvalues of A ⊗ B are the n2 products of the eigenvalues of A by the eigenvalues of B.Thus if the eigenvalues of A and B are,
respectively, ordered by:
(4.10)
δ1 ≥ δ2 ≥ · · · ≥ δn > 0,
η1 ≥ η2 ≥ · · · ≥ ηn > 0,
then in all the previous results in this section M = δ1 η1 and m = δn ηn . Thus
Eq (4.6) to Eq (4.9) become:
(4.11)
A2 ◦ B 2 ≤
(δ1 η1 + δn ηn )2
{A ◦ B}2
4δ1 η1 δn ηn
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
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(4.12)
(4.13)
A−1 ◦ B −1
(δ1 η1 + δn ηn )2
≤
{A ◦ B}−1
4δ1 η1 δn ηn
1
(A2 ◦ B 2 ) − (A ◦ B)2 ≤ (δ1 η1 − δn ηn )2 {I}
4
√
(4.14)
A−1 ◦ B
−1
− (A ◦ B)−1 ≤
√
δ1 η1 − δn ηn
{I} .
δ1 η1 δn ηn
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Corollary 4.4. Let A and B be a n×n positive definite Hermitian matrices. Let
m1 and M1 be, respectively, the smallest and the largest eigenvalues of A ⊗ I
and m2 and M2 , respectively, the smallest and the largest eigenvalues of I ⊗ B.
Then
(a) For p a nonzero integer, we have
(4.15)
Ap • B p ≤ max {λ1 , λ2 } (A • B)p ,
where λ1 and λ2 are given by Eq (3.31) and Eq (3.32).
While, for 0 < p < 1, we have the reverse inequality in Eq (4.15).
(b) For p a nonzero integer, we have
(4.16)
(Ap • B p ) − (A • B)p ≤ max {α1 , α2 } I,
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
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where α1 and α2 are given by Eq (3.34) and Eq (3.35).
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While, for 0 < p < 1, we have the reverse inequality in Eq (4.16).
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Note that, the eigenvalues of A ⊗ I equal the eigenvalues of A and the eigenvalues of I ⊗ B equal the eigenvalues of B.
Corollary 4.5. Let A and B be n × n positive definite Hermitian matrices. Let
m and M be, respectively, the smallest and the largest eigenvalues of A ⊕ B.
Then
(a) For p a nonzero integer, we have
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(4.17)
Ap • B p ≤ λ(A • B)p ,
J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
where, λ is given by Eq (3.21).
While, for 0 < p < 1, we have the reverse inequality in Eq (4.17).
(b) For p a nonzero integer, we have
(Ap • B p ) − (A • B)p ≤ αI,
(4.18)
where, α is given by Eq (3.23).
While, for 0 < p < 1, we have the reverse inequality in Eq (4.18).
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Special cases include from Eq (4.17):
(2.1) For p = 2, we have
(4.19)
A2 • B 2 ≤
(M + m)2
{A • B}2
4M m
(2.2) For p = −1, we have
(4.20)
A−1 • B −1 ≤
Title Page
2
(M + m)
{A • B}−1
4M m
Similarly, special cases include from Eq (4.18):
(2.1) For p = 2, we have
(4.21)
1
(A2 • B 2 ) − (A • B)2 ≤ (M − m)2 I
4
(2.2) For p = −1, we have
(A−1 • B −1 ) − (A • B)−1 ≤
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√
(4.22)
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√
M− m
{I} .
Mm
Page 30 of 37
J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
We note that the eigenvalues of A ⊕ B are the n2 sums of the eigenvalues of A
by the eigenvalues of B. Thus if the eigenvalues of A and B are, respectively,
ordered by:
δ1 ≥ δ2 ≥ · · · ≥ δn > 0,
η1 ≥ η2 ≥ · · · ≥ ηn > 0,
then in all previous results of this section M = δ1 + η1 and m = δn + ηn . Thus
Eq(4.19) to Eq (4.22) become:
(δ1 + η1 + δn + ηn )2
{A • B}2 ,
4(δ1 + η1 )(δn + ηn )
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
(δ1 + η1 + δn + ηn )2
≤
{A • B}−1 ,
4(δ1 + η1 )(δn + ηn )
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(4.23)
A2 • B 2 ≤
(4.24)
−1
A
•B
−1
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(4.25)
1
(A2 • B 2 ) − (A • B)2 ≤ ((δ1 + η1 ) − (δn + ηn ))2 I,
4
√
(4.26)
(A
−1
−1
−1
• B ) − (A • B)
≤
√
δ1 + η1 − δn + ηn
I.
(δ1 + η1 )(δn + ηn )
Corollary 4.6. Let A ≥ 0 and B ≥ 0 be compatibly matrices. Then
(4.27)
(4.28)
(i) (A ⊕ B)(A ⊕ B)∗ ≥ AA∗ ⊕ BB ∗
(ii) (A ⊕ B)w ≥ Aw ⊕ B w ,
for any positive integer w.
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
Corollary 4.7. Let A and B be matrices of order m × m and n × n respectively.
Then
(4.29)
(a)
(4.30)
(b)
tr(A ⊕ B) = n · tr(A) + m · tr(B)
√
√
kA ⊕ Bkp ≤ p n kAkp + p m kBkp ,
where
(4.31)
(c)
kAkp = [tr |A|p ]1/p , 1 ≤ p < ∞.
eA⊕B = eA ⊗ eB
Corollary 4.8. Let A and B be non singular matrices of order m×m and n×n,
respectively. Then
(4.32)
(4.33)
(4.34)
(i) (A ⊕ B)−1 = (A−1 ⊕ B −1 )−1 (A−1 ⊗ B −1 )
(ii) (A ⊕ B)−1 = (A−1 ⊗ In )(A−1 ⊕ B −1 )−1 (Im ⊗ B −1 )
(iii) (A ⊕ B)−1 = (Im ⊗ B −1 )(A−1 ⊕ B −1 )−1 (A−1 ⊗ In )
In [1], Ando proved the following inequality;
(4.35)
1
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1
A ◦ B ≤ (Ap ◦ I) p (B q ◦ I) q ,
where A and B are positive definite matrices and p, q ≥ 1with 1/p + 1/q = 1.
If k·k is a unitarily invariant norm and k·k∞ is the spectral norm, Horn and
Johnson in [3] proved the following three conditions are equivalent:
(4.36)
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
(i) kAk∞ ≤ kAk
(ii) kABk ≤ kAk · kBk
(iii) kA ◦ Bk ≤ kAk · kBk
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J. Ineq. Pure and Appl. Math. 7(1) Art. 34, 2006
for all matrices A and B.
In [2], Hiai and Zhan proved the following inequalities:
(4.37)
kABk
kA + Bk
≤
kAk · kBk
kAk + kBk
and
kA ◦ Bk
kA + Bk
≤
kAk · kBk
kAk + kBk
for any invariant norm with kdiag(1, 0, . . ., 0)k ≥ 1 and A, B are nonzero positive definite matrices.
We have an application to generalize the inequalities in Eq (4.37) involving
the Hadamard product (sum) and the Kronecker product (sum).
Theorem 4.9. Let k·k be a unitarily invariant norm with kdiag(1, 0, . . . , 0)k ≥
1 and A and B be nonzero positive definite matrices. Then
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
kA • Bk
kA ◦ Bk
≤
.
kAk · kBk
kAk + kBk
Zeyad Al Zhour and
Adem Kilicman
Proof. Let k·k∞ be the spectral norm and applying Eq (4.35) to A/ kAk∞ ≤ I,
B/ kBk∞ ≤ I and using the Young inequality for scalars, we get
p p1 q 1q
A
B
A
B
◦
≤
◦I
◦I
kAk∞
kBk∞
kAk∞
kBk∞
p
q
A
1
B
1
◦I +
◦I
≤
p kAk∞
q kBk∞
1
A
1
B
◦I +
◦I
≤
p kAk∞
q kBk∞
1
A
1
B
=
+
◦I
p kAk∞
q kBk∞
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(4.38)
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We choose
kAk∞
1
=
p
[kAk∞ + kBk∞ ]
and
kBk∞
1
=
.
q
[kAk∞ + kBk∞ ]
Since kAk∞ ≤ kAk and kBk∞ ≤ kBk thanks tokdiag(1, 0, . . . , 0)k ≥ 1, we
obtain
kAk∞ · kBk∞
(4.39)
A◦B ≤
(A + B) ◦ I
kAk∞ + kBk∞
kAk · kBk
≤
(A • B)
kAk + kBk
Zeyad Al Zhour and
Adem Kilicman
Hence,
kA ◦ Bk ≤
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
kAk · kBk
kA • Bk
kAk + kBk
or
kA ◦ Bk
kA • Bk
≤
kAk · kBk
kAk + kBk
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Corollary 4.10. Let k·k be a unitarily invariant norm with kdiag(1, 0, . . . , 0)k ≥
1 and A and B be nonzero positive definite matrices. Then
(4.40)
kA ⊗ Bk
kA ⊕ Bk
≤
.
kAk · kBk
kAk + kBk
Proof. Applying Eq (2.7) and Eq (4.39), we have
K 0 (A ⊗ B)K ≤
kAk · kBk 0
K (A ⊕ B)K
kAk + kBk
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and
kK 0 (A ⊗ B)Kk ≤
kAk · kBk
kK 0 (A ⊕ B)Kk .
kAk + kBk
Provided that k·k is unitarily invariant norm, we get the result.
Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
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References
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Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
Adem Kilicman
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Close
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Matrix Equalities and
Inequalities Involving
Khatri-Rao and Tracy-Singh
Sums
Zeyad Al Zhour and
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