HADAMARD PRODUCT VERSIONS OF THE
CHEBYSHEV AND KANTOROVICH INEQUALITIES
JAGJIT SINGH MATHARU AND JASPAL SINGH AUJLA
Department of Mathematics
National Institute of Technology
Jalandhar 144011, Punjab, INDIA
EMail: matharujs@yahoo.com
aujlajs@nitj.ac.in
Hadamard Product Versions
Jagjit Singh Matharu and
Jaspal Singh Aujla
vol. 10, iss. 2, art. 51, 2009
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15 April, 2009
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Communicated by:
S. Puntanen
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2000 AMS Sub. Class.:
Primary 15A48; Secondary 15A18, 15A45.
Key words:
Chebyshev inequality, Kantorovich inequality, Hadamard product
Abstract:
The purpose of this note is to prove Hadamard product versions of the Chebyshev
and the Kantorovich inequalities for positive real numbers. We also prove a
generalization of Fiedler’s inequality.
Received:
10 February, 2009
Accepted:
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Acknowledgements:
The authors thank a referee for useful suggestions.
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Contents
1
Introduction
3
2
The Chebyshev and Kantorovich Inequalities: Matrix Versions
5
Hadamard Product Versions
Jagjit Singh Matharu and
Jaspal Singh Aujla
vol. 10, iss. 2, art. 51, 2009
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1.
Introduction
In what follows, the capital letters A, B, C, . . . denote m × m complex matrices,
whereas the small letters a, b, c, . . . denote real numbers, unless mentioned otherwise. By X ≥ Y we mean that X − Y is positive semidefinite (X > Y mean
X − Y is positive definite). For A = (aij ) and B = (bij ), A ◦ B = (aij bij ) denotes the Hadamard product of A and B. According to Schur’s theorem [4, Page
23] the Hadamard product is monotone in the sense that A ≥ B, C ≥ D implies
A ◦ C ≥ B ◦ D. The tensor product A ⊗ B is the m2 × m2 matrix
Hadamard Product Versions
Jagjit Singh Matharu and
Jaspal Singh Aujla
vol. 10, iss. 2, art. 51, 2009
a11 B
 ...

(1.1)
···

a1m B
..
.
.
am1 B · · · amm B
Marcus and Khan in [10] made the simple but important observation that the Hadamard
product is a principal submatrix of the tensor product. The inequality
! n
!
n
n
X
X
X
(1.2)
wi ai
wi bi ≤
wi ai bi
i=1
i=1
i=1
holds for all a1 ≥ a2 ≥ · · · ≥ an ≥ 0, b1 ≥ b2 ≥ · · · ≥ bn ≥ 0 and weights
wi ≥ 0, i = 1, . . . , n. Hardy, Littlewood and Polya [6, page 43] attribute this
inequality to Chebyshev. For 0 < a ≤ ai ≤ b, wi ≥ 0, i = 1, 2, . . . , n, Kantorovich’s
inequality states that
! n
!
!2
n
n
X
X wi
(a + b)2 X
(1.3)
wi ai
≤
wi .
a
4ab
i
i=1
i=1
i=1
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In Section 2, we state and prove matrix versions of inequalities (1.2) and (1.3) involving the Hadamard product. A generalization of Fiedler’s inequality is also proved in
this section. There are several generalizations of Kantorovich and Fiedler’s inequality; see [2, 3, 8, 9].
Hadamard Product Versions
Jagjit Singh Matharu and
Jaspal Singh Aujla
vol. 10, iss. 2, art. 51, 2009
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2.
The Chebyshev and Kantorovich Inequalities:
Matrix Versions
We begin with a Hadamard product version of inequality (1.2).
Theorem 2.1. Let A1 ≥ · · · ≥ An ≥ 0 and B1 ≥ · · · ≥ Bn ≥ 0. Then
!
!
! n
!
n
n
n
X
X
X
X
(2.1)
wi Ai ◦
wi Bi ≤
wi
wi (Ai ◦ Bi ) ,
i=1
i=1
i=1
i=1
vol. 10, iss. 2, art. 51, 2009
where wi ≥ 0, i = 1, . . . , n, are weights.
Proof. We have
(2.2)
n
X
Hadamard Product Versions
Jagjit Singh Matharu and
Jaspal Singh Aujla
Title Page
!
wi
n
X
i=1
=
=
!
wi (Ai ◦ Bi )
i=1
n
X
−
n
X
!
wi Ai
◦
i=1
n
X
!
i=1
(wi wj (Aj ◦ Bj ) − wi wj (Ai ◦ Bj ))
i,j=1
n
X
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wi Bi
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1
wi wj (Aj ◦ Bj ) − wi wj (Ai ◦ Bj )
2 i,j=1
+ wj wi (Ai ◦ Bi ) − wj wi (Aj ◦ Bi )
n
1X
wi wj (Ai − Aj ) ◦ (Bi − Bj ).
=
2 i,j=1
Since the Hadamard product of two positive semidefinite matrices is positive semidefinite, therefore the summand in 2.2 is positive semidefinite.
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Our next result is a Hadamard product version of inequality (1.3) .
Theorem 2.2. Let A1 , . . . , An be such that 0 < aIm ≤ Ai ≤ bIm , i = 1, . . . , n
(here Im denotes the m × m identity matrix). Then
!
!
n
n
X
X
1/2
1/2
1/2 −1
1/2
W i Ai W i
◦
W i Ai W i
(2.3)
i=1
i=1
2
≤
a +b
2ab
2
n
X
!
Wi
n
X
◦
i=1
!
Wi
i=1
Hadamard Product Versions
Jagjit Singh Matharu and
Jaspal Singh Aujla
vol. 10, iss. 2, art. 51, 2009
for all Wi ≥ 0, i = 1, . . . , n.
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Proof. We first prove the inequality
Contents
(2.4) P 1/2 AP 1/2 ◦ Q1/2 B −1 Q1/2 + P 1/2 A−1 P 1/2 ◦ Q1/2 BQ1/2 ≤
2
2
a +b
(P ◦ Q),
ab
when 0 < aIm ≤ A, B ≤ bIm and P, Q ≥ 0. Let A = U DU ∗ and B = V ΓV ∗ with
unitary U and V , and diagonal matrices D and Γ. Then
A ⊗ B −1 + A−1 ⊗ B = (U ⊗ V )(D ⊗ Γ + Γ−1 ⊗ D)(U ⊗ V )∗
2
a + b2
≤ (U ⊗ V )
(Im ⊗ Im ) (U ⊗ V )∗
ab
2
2
a +b
=
(Im ⊗ Im ),
ab
where the inequality follows from (1.3). Thus we have
(2.5)
P 1/2 AP 1/2 ⊗ Q1/2 B −1 Q1/2 + P 1/2 A−1 P 1/2 ⊗ Q1/2 BQ1/2
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= (P 1/2 ⊗ Q1/2 )(A ⊗ B −1 + A−1 ⊗ B)(P 1/2 ⊗ Q1/2 )
a2 + b2
(P ⊗ Q).
≤
ab
Since the Hadamard product is a principal submatrix of the tensor product, the inequality (2.4) follows from (2.5). On taking B = A and Q = P in (2.4) we see that
(2.3) holds for n = 1. Further, by (2.4) we have
1/2
1/2
W i Ai W i
1/2
1/2
◦Wj A−1
j Wj
1/2
1/2
+Wi A−1
i Wi
1/2
1/2
◦Wj Aj Wj
a2 + b 2
(Wi ◦Wj )
≤
ab
Hadamard Product Versions
Jagjit Singh Matharu and
Jaspal Singh Aujla
vol. 10, iss. 2, art. 51, 2009
for i, j = 1, . . . , n. Summing over i, j, we have
(2.6) 2
n h
X
1/2
1/2
W i Ai W i
i,j=1
n
i a2 + b2 X
1/2 −1
1/2
(Wi ◦ Wj ),
◦ W j Aj W j
≤
ab
i,j=1
which implies that
n
X
i=1
!
1/2
1/2
W i Ai W i
◦
n
X
i=1
!
1/2
1/2
Wi A−1
i Wi
≤
a2 + b 2
2ab
X
n
i=1
!
Wi ◦
n
X
!
Wi .
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Page 7 of 13
i=1
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The next corollary follows on taking Wi = wi Im , i = 1, . . . , n.
Corollary 2.3. Let A1 , . . . , An be such that 0 < aIm ≤ Ai ≤ bIm , and wi ≥ 0, i =
1, . . . , n be weights. Then
!
! !2
X
n
n
n
2
2
X
X
a +b
wi Ai ◦
wi A−1
≤
wi Im .
i
2ab
i=1
i=1
i=1
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Remark 1. The case n = 1 of Corollary 2.3 is proved in [7]. The example
√
√
3− 5
3+ 5
2 1
A=
, a=
, b=
1 1
2
2
shows that the inequality
A ◦ A−1 ≤
(a + b)2
I2
4ab
Hadamard Product Versions
Jagjit Singh Matharu and
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need not be true.
vol. 10, iss. 2, art. 51, 2009
For our next result we need the following lemma.
Lemma 2.4. Let 0 ≤ r ≤ 1. Then Ar + A−r ≤ A + A−1 for all A > 0.
Proof. Suppose that A = U ΓU ∗ with unitary U and diagonal matrix Γ. Then
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r
A +A
−r
r
−r
= U (Γ + Γ )U
∗
≤ U (Γ + Γ−1 )U ∗ = A + A−1
since xr + x−r ≤ x + x−1 for any positive real number x and 0 ≤ r ≤ 1.
≤
i=1
n
X
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Theorem 2.5. Let 0 ≤ α < β. Then
!
!
n
n
X
X
1/2
1/2
1/2
1/2
Wi Aαi Wi
◦
Wi A−α
i Wi
i=1
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!
1/2
1/2
Wi Aβi Wi
i=1
for all Ai > 0 and Wi ≥ 0, i = 1, . . . , n.
◦
n
X
i=1
!
Wi A−β
i Wi
1/2
1/2
Proof. We first prove the inequality
1/2
1/2
1/2
1/2
(2.7)
Wi Aαi Wi
◦ Wj A−α
W
j
j
1/2 −α
1/2
1/2 α
1/2
+ W i Ai W i
◦ W j Aj W j
1/2
1/2
1/2
1/2
W
◦ Wj A−β
≤ Wi Aβi Wi
j
j
1/2
1/2
1/2 β
1/2
+ Wi A−β
W
◦
W
A
W
i
i
j
j
j
Hadamard Product Versions
Jagjit Singh Matharu and
Jaspal Singh Aujla
vol. 10, iss. 2, art. 51, 2009
for 0 ≤ α < β. Let 0 ≤ r ≤ 1. Then
1/2
1/2
1/2
1/2
1/2 −r
1/2
1/2 r
1/2
Wi Ari Wi
⊗ Wj A−r
W
+
W
A
W
⊗
W
A
W
j
j
j
i
i
i
j
j
1/2
1/2
1/2
1/2
−r
r
= Wi ⊗ Wj
Ari ⊗ A−r
Wi ⊗ Wj
j + Ai ⊗ Aj
1/2
1/2
1/2
1/2
−1 r
−1 −r
= Wi ⊗ Wj
Wi ⊗ Wj
Ai ⊗ Aj
+ A i ⊗ Aj
1/2
1/2
1/2
1/2
−1 −1
≤ Wi ⊗ Wj
Ai ⊗ A−1
+
A
⊗
A
W
⊗
W
i
j
j
i
j
where the inequality follows from Lemma 2.4. Taking r = α/β and replacing Ai by
Aβi and Aj by Aβj , we have
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
Wi Aαi Wi
⊗ Wj A−α
+ Wi A−α
⊗ Wj Aαj Wj
j Wj
i Wi
1/2
1/2
1/2
1/2
1/2 −β
1/2
1/2 β
1/2
≤ Wi Aβi Wi
⊗ Wj A−β
W
+
W
A
W
⊗
W
A
W
.
j
j
i
i
i
j
j
j
Again using the fact that the Hadamard product is a principal submatrix of the tensor
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product, the preceding inequality implies (2.7). Summing over i, j in (2.7), we have
!
!
n
n
X
X
1/2 α
1/2
1/2 −α
1/2
W i Ai W i
◦
W i Ai W i
i=1
i=1
≤
n
X
!
1/2
1/2
Wi Aβi Wi
◦
i=1
n
X
!
1/2
1/2
Wi A−β
i Wi
.
Hadamard Product Versions
Jagjit Singh Matharu and
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i=1
vol. 10, iss. 2, art. 51, 2009
Corollary 2.6. Let 0 ≤ α < β. Then
!
!
n
n
X
X
Aαi ◦
A−α
≤
j
i=1
n
X
j=1
!
Aβi
i=1
◦
n
X
!
A−β
j
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j=1
for all Ai > 0, i = 1, . . . , n.
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Proof. Taking Wi = Im in Theorem 2.5 we get the desired result.
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Corollary 2.7. Let 0 ≤ β. Then
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Im ≤
n
X
!
1/2
1/2
Wi Aβi Wi
◦
i=1
n
X
!
1/2
1/2
Wi A−β
i Wi
i=1
for all Ai > 0 and Wi ≥ 0, i = 1, . . . , n, where
Pn
i=1
Wi = Im .
Proof. Taking α = 0 in Theorem 2.5 gives the desired inequality.
Remark 2. Corollary 2.7 is another generalization of Fiedler’s inequality [5]
A ◦ A−1 ≥ Im .
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Next we prove a convexity theorem involving the Hadamard product.
Theorem 2.8. The function
f (t) = A1+t ◦ B 1−t + A1−t ◦ B 1+t
is convex on the interval [−1, 1] and attains its minimum at t = 0 for all A, B > 0.
Proof. Since f is continuous we need to prove only that f is mid-point convex. Note
that for A, B > 0 and s, t in [−1, 1] the matrices
1+s+t
1−(s+t)
A
A1+s
A
A1−s
,
,
A1+s A1+(s−t)
A1−s
A1−(s−t)
1+s+t
1−(s+t)
B
B 1+s
B
B 1−s
,
B 1+s B 1+(s−t)
B 1−s
B 1−(s−t)
are positive semidefinite. Hence the matrix
1+s+t
A
◦ B 1−(s+t) + A1−(s+t) ◦ B 1+s+t
A1+s ◦ B 1−s + A1−s ◦ B 1+s
X=
A1+s ◦ B 1−s + A1−s ◦ B 1+s
A1+(s−t) ◦ B 1−(s−t) + A1−(s−t) ◦ B 1+(s−t)
is positive semidefinite. Similarly, the matrix
1+(s−t)
A
◦ B 1−(s−t) + A1−(s−t) ◦ B 1+(s−t)
A1+s ◦ B 1−s + A1−s ◦ B 1+s
Y=
A1+s ◦ B 1−s + A1−s ◦ B 1+s
A1+(s+t) ◦ B 1−(s+t) + A1−(s+t) ◦ B 1+s+t
is positive semidefinite. Hence
f (s + t) + f (s − t)
2f (s)
(2.8)
X +Y =
2f (s)
f (s + t) + f (s − t)
is positive semidefinite, which implies that
1
f (s) ≤ [f (s + t) + f (s − t)].
2
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This proves the convexity of f . Further, note that f (t) = f (−t). This together with
the convexity of f implies that f attains its minimum at 0.
Corollary 2.9. The function
g(t) = At ◦ B 1−t + A1−t ◦ B t
is decreasing on [0, 1/2], increasing on [1/2, 1], and attains its minimum at t =
for all A, B > 0.
Proof. The proof follows on replacing A, B by A1/2 , B 1/2 and t by
2.8.
1+t
2
1
2
in Theorem
A norm |||·||| on m×m complex matrices is called unitarily invariant if |||U XV ||| =
|||X||| for all unitary matrices U, V . If A is positive semidefinite and X is any matrix,
then
|||A ◦ X||| ≤ max aii |||X|||
for all unitarily invariant norms ||| · ||| [1]. Thus the proof of the following corollary
follows from Corollary 2.9 using the fact that g(1/2) ≤ g(t) ≤ g(1) = g(0).
Corollary 2.10. Let 0 ≤ t ≤ 1. Then,
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2|||A1/2 ◦ B 1/2 ||| ≤ |||At ◦ B 1−t + A1−t ◦ B t ||| ≤ |||A + B|||
for all unitarily invariant norms ||| · ||| and all A, B > 0.
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References
[1] T. ANDO, R.A. HORN AND C.R. JOHNSON, The singular values of the
Hadamard product: A basic inequality, Linear Multilinear Algebra, 21 (1987),
345–365.
[2] J.K. BAKSALARY AND S. PUNTANEN, Generalized matrix versions of the
Cauchy-Schwarz and Kantorovich inequalities, Aequationes Math., 41 (1991),
103–110.
[3] R.B. BAPAT AND M.K. KWONG, A generalisation of A ◦ A−1 ≥ I, Linear
Algebra Appl., 93 (1987), 107–112.
[4] R. BHATIA, Matrix Analysis, Springer Verlag, New York, 1997.
Hadamard Product Versions
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vol. 10, iss. 2, art. 51, 2009
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[5] M. FIEDLER, Über eine Ungleichung für positiv definite Matrizen, Math.
Nachrichten, 23 (1961), 197–199.
[6] G.H. HARDY, J.E. LITTLEWOOD
University Press, Cambridge, 1959.
AND
G. POLYA, Inequalities, Cambridge
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[7] J. MIĆIĆ, J. PECARIC AND Y. SEO, Complementary inequalities to inequalities of Jensen and Ando based on the Mond-Pečarić method, Linear Algebra
Appl., 318 (2000), 87–107.
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[8] A.W. MARSHALL AND I. OLKIN, Matrix versions of the Cauchy and Kantorovich inequalities, Aequationes Math., 40 (1990), 89–93.
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[9] M. SINGH, J.S. AUJLA AND H.L. VASUDEVA, Inequalities for Hadamard
product and unitarily invariant norms of matrices, Linear Multilinear Algebra,
48 (2000), 247–262.
[10] M. MARCUS AND N.A. KHAN, A note on Hadamard product, Canad. Math.
Bull., 2 (1950), 81–83.
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