Volume 10 (2009), Issue 2, Article 51, 6 pp.
HADAMARD PRODUCT VERSIONS OF THE CHEBYSHEV AND
KANTOROVICH INEQUALITIES
JAGJIT SINGH MATHARU AND JASPAL SINGH AUJLA
D EPARTMENT OF M ATHEMATICS
NATIONAL I NSTITUTE OF T ECHNOLOGY
JALANDHAR 144011, P UNJAB , INDIA
matharujs@yahoo.com
aujlajs@nitj.ac.in
Received 10 February, 2009; accepted 15 April, 2009
Communicated by S. Puntanen
A BSTRACT. 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.
Key words and phrases: Chebyshev inequality, Kantorovich inequality, Hadamard product.
2000 Mathematics Subject Classification. Primary 15A48; Secondary 15A18, 15A45.
1. I NTRODUCTION
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
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
The authors thank a referee for useful suggestions.
043-09
i=1
i=1
2
JAGJIT S INGH M ATHARU
AND JASPAL
S INGH AUJLA
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
!
!2
! n
n
n
X
X wi
(a + b)2 X
(1.3)
wi ai
≤
wi .
ai
4ab
i=1
i=1
i=1
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].
2. T HE C HEBYSHEV AND K ANTOROVICH I NEQUALITIES : M ATRIX V ERSIONS
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
where wi ≥ 0, i = 1, . . . , n, are weights.
Proof. We have
(2.2)
n
X
=
=
!
wi
i=1
n
X
n
X
!
wi (Ai ◦ Bi )
i=1
−
n
X
!
wi Ai
i=1
◦
n
X
!
wi Bi
i=1
(wi wj (Aj ◦ Bj ) − wi wj (Ai ◦ Bj ))
i,j=1
n
X
1
wi wj (Aj ◦ Bj ) − wi wj (Ai ◦ Bj ) + wj wi (Ai ◦ Bi ) − wj wi (Aj ◦ Bi )
2 i,j=1
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.
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
n
n
2
2
X
X
X
X
a
+
b
1/2
1/2
1/2
1/2
(2.3)
W i Ai W i
◦
Wi A−1
≤
Wi ◦
Wi
i Wi
2ab
i=1
i=1
i=1
i=1
for all Wi ≥ 0, i = 1, . . . , n.
Proof. We first prove the inequality
(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 ≤
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 51, 6 pp.
a2 + b 2
(P ◦ Q),
ab
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H ADAMARD P RODUCT V ERSIONS
3
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
a2 + b 2
=
(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
= (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
◦ W j Aj W j
≤
a2 + b 2
(Wi ◦ Wj )
ab
for i, j = 1, . . . , n. Summing over i, j, we have
n
n h
i a2 + b2 X
X
1/2
1/2
1/2 −1
1/2
(2.6)
2
W i Ai W i
◦ W j Aj W j
≤
(Wi ◦ Wj ),
ab
i,j=1
i,j=1
which implies that
n
X
!
1/2
1/2
W i Ai W i
i=1
◦
n
X
!
1/2
1/2
Wi A−1
i Wi
≤
i=1
a2 + b 2
2ab
X
n
!
Wi
i=1
◦
n
X
!
Wi .
i=1
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 Im .
wi A−1
≤
i
2ab
i=1
i=1
i=1
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
need not be true.
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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 51, 6 pp.
http://jipam.vu.edu.au/
4
JAGJIT S INGH M ATHARU
AND JASPAL
S INGH AUJLA
Proof. Suppose that A = U ΓU ∗ with unitary U and diagonal matrix Γ. Then
Ar + A−r = U (Γr + Γ−r )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.
Theorem 2.5. Let 0 ≤ α < β. Then
!
!
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
!
Wi A−β
i Wi
1/2
1/2
i=1
for all Ai > 0 and Wi ≥ 0, i = 1, . . . , n.
Proof. We first prove the inequality
1/2
1/2
1/2
1/2
1/2 −α
1/2
1/2 α
1/2
◦
W
A
W
(2.7)
Wi Aαi Wi
◦ Wj A−α
W
+
W
A
W
j
j
j
i
i
i
j
j
1/2 β
1/2
1/2 −β
1/2
1/2 −β
1/2
1/2 β
1/2
≤ W i Ai W i
◦ W j Aj W j
+ W i Ai W i
◦ W j Aj W j
for 0 ≤ α < β. Let 0 ≤ r ≤ 1. Then
1/2 r
1/2
1/2 −r
1/2
1/2 −r
1/2
1/2 r
1/2
W i Ai W i
⊗ W j Aj W j
+ W i Ai W i
⊗ W j Aj W j
1/2
1/2
1/2
1/2
−r
r
+
A
⊗
A
W
⊗
W
= Wi ⊗ Wj
Ari ⊗ A−r
j
j
i
i
j
1/2
1/2
1/2
1/2
−1 r
−1 −r
= Wi ⊗ Wj
Ai ⊗ A j
+ Ai ⊗ A j
Wi ⊗ Wj
1/2
1/2
1/2
1/2
−1 −1
≤ Wi ⊗ Wj
W
Ai ⊗ A−1
+
A
⊗
A
⊗
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 −α
1/2
1/2 α
1/2
Wi Aαi Wi
⊗ Wj A−α
W
+
W
A
W
⊗
W
A
W
j
j
j
i
i
i
j
j
1/2 β
1/2
1/2 −β
1/2
1/2 −β
1/2
1/2
1/2 β
.
⊗ W j Aj W j
+ W i Ai W i
⊗ W j Aj W j
≤ W i Ai W i
Again using the fact that the Hadamard product is a principal submatrix of the tensor 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
◦
n
X
!
1/2
1/2
Wi A−β
i Wi
.
i=1
i=1
Corollary 2.6. Let 0 ≤ α < β. Then
!
!
n
n
X
X
Aαi ◦
A−α
≤
j
i=1
j=1
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 51, 6 pp.
n
X
i=1
!
Aβi
◦
n
X
!
A−β
j
j=1
http://jipam.vu.edu.au/
H ADAMARD P RODUCT V ERSIONS
5
for all Ai > 0, i = 1, . . . , n.
Proof. Taking Wi = Im in Theorem 2.5 we get the desired result.
Corollary 2.7. Let 0 ≤ β. Then
!
n
X
1/2 β
1/2
Im ≤
W i Ai W i
◦
n
X
i=1
!
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 .
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
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
1
2
for all
in Theorem 2.8.
is decreasing on [0, 1/2], increasing on [1/2, 1], and attains its minimum at t =
A, B > 0.
Proof. The proof follows on replacing A, B by A1/2 , B 1/2 and t by
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 51, 6 pp.
1+t
2
http://jipam.vu.edu.au/
6
JAGJIT S INGH M ATHARU
AND JASPAL
S INGH AUJLA
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,
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.
R EFERENCES
[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.
[5] M. FIEDLER, Über eine Ungleichung für positiv definite Matrizen, Math. Nachrichten, 23 (1961),
197–199.
[6] G.H. HARDY, J.E. LITTLEWOOD
Cambridge, 1959.
AND
G. POLYA, Inequalities, Cambridge University Press,
[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.
[8] A.W. MARSHALL AND I. OLKIN, Matrix versions of the Cauchy and Kantorovich inequalities,
Aequationes Math., 40 (1990), 89–93.
[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
81–83.
AND
N.A. KHAN, A note on Hadamard product, Canad. Math. Bull., 2 (1950),
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 51, 6 pp.
http://jipam.vu.edu.au/