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The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society.
Volume 6, pp. 56-61, March 2000.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela

ELA

ON INEQUALITIES INVOLVING THE
HADAMARD PRODUCT OF MATRICES




z
B. MONDy AND J. PECARI
C

Abstra
t. Re
ently, the authors established a number of inequalities involving integer powers of
the Hadamard produ

t of two positive denite Hermitian matri
es. Here these results are extended
in two ways. First, the restri
tion to integer powers is relaxed to in
lude all real numbers not in
the open interval ( 1; 1). Se
ond, the results are extended to the Hadamard produ
t of any nite
number of Hermitian positive denite matri
es.
Key words. matrix inequalities, Hadamard produ
t
AMS subje
t
lassi
ations. 15A45
1. Introdu
tion. Let A and B be n  n matri
es. A Æ B denotes the Hadamard
produ

t and A
B the Krone
ker produ
t of A and B .
These two produ
ts are related by the following relation [2℄, [3℄.
There exists an n2  n sele
tion matrix J su
h that J T J = I and
A

Æ

B

= J T (A
B )J:

Note that J T is the n  n2 matrix [E11 E22 : : : Enn ℄, where Eii is the n  n matrix of
zeros ex

ept for a one in the (i; i)th position.
Using this result, in [4℄ the authors proved a number of inequalities involving
integer powers of the Hadamard produ
t of two positive denite Hermitian matri
es.
Here we extend these results in two ways. First, the restri
tion to integer powers
is relaxed to in
lude all real numbers not in the open interval ( 1; 1). Se
ond, the
results are extended to the Hadamard produ
t of any nite number of n  n Hermitian
positive denite matri
es.
2. Notation and Preliminary Results.

u
ts of matri
es Ai ;

i

The Hadamard and Krone
ker prod-

= 1; : : : k , will be denoted by

k

Æ
i=1

and

k
O
i=1

, respe
tively.

We shall make frequent use of the following property of the Krone
ker produ
t:
(AB )
(C D) = (A
C )(B
D):

For a nite number of matri
es Ai ;
(1)

!
k
Y
A

i=1

i




i
k

B ; i

= 1; : : : ; k , this be
omes

k
Y ! Y
B

i=1

i

=

i=1

(Ai
Bi ):

 Re
eived by the editors on 11 O
tober 1999. A

epted for publi
ation on 25 January 2000.
Handling editor: Daniel Hershkowitz.
y Department of Mathemati
s, La Trobe University, Bundoora, Vi
toria 3083, Australia
(matbmlura
.latrobe.edu.au).

z Fa
ulty of Textile Te
hnology, University of Zagreb, Zagreb, Croatia, and Applied
Mathemati
s Department, University of Adelaide, Adelaide, South Australia 5005, Australia
(jpe
ari
maths.adelaide.edu.au).

56

ELA

On Inequalities Involving the Hadamard Produ
t of Matri
es

57

Let A be a positive denite n  n Hermitian matrix. There exists a matrix U su

h
that
A = U  [1 ; 2 ; : : : ; n ℄U; U  U = I;

where [1 ; 2 ; : : : ; n ℄ is the diagonal matrix with i , the positive eigenvalues of A,
along the diagonal [1℄. For any real number s; As is dened by
As = U  [s1 ; s2 ; : : : ; sn ℄U:

Lemma 2.1. Let A and B be positive denite Hermitian n  n matri
es and s a
nonzero real number. Then

As
B s = (A
B )s :

(2)

Proof. Assume
B = V  [

1 ;
2 ; : : : ;
n ℄V; V  V = I;

where
i are the eigenvalues of B . Then
As
B s

= (U  [s1 ; : : : ; sn ℄U )
(V  [
1s ; : : : ;
ns ℄V )
= (U 
V  )([s1 ; : : : ; sn ℄
[
1s ; : : : ;
ns ℄)(U
V )
= (U

V ) ([s1 ; : : : ; sn ℄
[
1s ; : : : ;
ns ℄)(U
V ) = (A
B )s :

Note that
(U


V ) (U
V )

= (U 
V  )(U
V )
= (U  U )
(V  V ) = I
I = In2 :

Equation (2) extends readily, for a nite number of n  n positive denite Hermitian
matri
es Ai ; i = 1; : : : ; k , to
k
O

(3)

i=1

(Asi ) =

!s
k
O
i=1

Ai

:

Let Ai ; i = 1; : : : ; k , be n  n matri
es. There exists an nk
sele
tion matrix P su
h that P T P = I
and
Lemma 2.2.

k

Æ A = PT
i=1 i

(4)

!
k
O
i=1

Ai P:

We prove this for three matri
es.1 The extension from m to m + 1 is similar.
AÆBÆC

1 This

= A Æ J T (B
C )J
= J T (A
(J T (B
C )J ))J
= J T ((IAI )
(J T (B
C )J ))J
= J T (I
J T )(A
B
C )(I
J )J
= J T (I
J )T (A
B
C )(I
J )J
= J^T (A
B
C )J^;

proof was provided by George Visi
k in a private
ommuni
ation.

by (1)

n

ELA
58

B. Mond and J. Pe
ari

where J^ is the

3

n



matrix J^ = (I
J )J . Note that J^T J^ = I .

n

3. Results. In this se
tion, Ai ; i = 1; : : : ; k , will denote n  n positive denite
Hermitian matri
es. Ai  Aj means that Ai Aj is positive semidenite.
Theorem 3.1.
Let r and s be real numbers r < s, and either r 2
= ( 1; 1) and
1
s 2
= ( 1; 1) or s  1  r 
2
or r 
1  s  21 . Then



(5)

Æk

i=1

s

1=s





Ai

Æk

i=1

r

1=r

:

Ai

We make use of the following result [5℄.
Let A be an n  n positive denite Hermitian matrix and let V be an n  t matrix
su
h that V  V = I . Then
Proof.

(V  As V )1=s

 (  r )1=r
, su
h that either 2 ( 1 1) and 2 (
V

A V

for all real r and s, r < s
r =
;
s =
1; 1) or s  1  r  21
1
or r  1  s  2 .
Here instead of V , we use the nk  n sele
tion matrix P given by (4). Noting (3),
we have



=
=

Æk

s

1=s

 i=1Nk


P

P

t



=

Ai

i=1 Ai


t Nk

P

s

r
i=1 Ai

t

P

 1=s
Nk
s
i=1 Ai P
1=s  Nk



 1=r

=

P



k

Æ

i=1

or, equivalently







Æk

Æk



i=1

1

Ai

i=1

or, equivalently,

1



Æk

i=1

For r > 1, we have

Ai


Ai

A

1=r
i

1





k

Æ







Æk

Æk



Æk

A

Æk

i=1



Ai





i=1

Æk

i=1

1


:

1=r

Ai

Ai

For r = 2, the last two inequalities be
ome



Ai

i

r

:



Æk

i=1

i=1



r

i=1 Ai
1=r

Ai

i=1





T

i=1

Some spe
ial
ases of (5) are the following:



P

2

Ai

1=r

1=2

:

r 1=r
P

ELA
On Inequalities Involving the Hadamard Produ
t of Matri
es
and


Theorem 3.2.

s2
= ( 1; 1)

or

Let

r2
= ( 1; 1).

r

Æk A1=2
i=1 i
s

and







Æk Ai
i=1

1=2

59

:

be nonzero real numbers su
h that

s > r

and

Then



(a)

Æk As
i=1 i

1=s

 4



Æk Ar
i=1 i

1=r

;

where

= M=m,

k
O



4 =

(6)

Ai .

and

M

r(
s
r )
(s r)(
r 1)

and

m

1=s 

s(
r
s )
(r s)(
s 1)



1=r

;

are, respe
tively, the largest and smallest eigenvalues of

Also,

i=1



(b)

Æk As
i=1 i



1=s

Æk Ar
i=1 i

1=r

 4I;

where

4 = max
f[M s + (1
2[0;1℄

(7)

)ms ℄1=s

)mr ℄1=r g:

[M r + (1

Proof. Let A be an n  n positive denite Hermitian matrix with eigenvalues
ontained in the interval [m; M ℄, where 0 < m < M , and let V be an n  t matrix
su
h that V  V = I . If r and s are nonzero real numbers su
h that r < s and either
s2
= ( 1; 1) or r 2
= ( 1; 1), then [6℄

(V  As V )1=s

(8)

 4 (V  Ar V )1=r

 is given by (6), and
where 4
(V  As V )1=s

(9)

(V  Ar V )1=r

 4I

where 4 is given by (7). Thus for part (a), from (8) and noting (3) and (4), we have



 4

"
PT

k

Æ

As
i=1 i

k
O
i=1

1=s

"
= P

!r #1=r
Ai

P

T

k
O
i=1

"

! #1=s
Asi

 PT
=4

P
k
O
i=1

"
= P

T

! #1=r
Ari P

k
O

!s #1=s
Ai

P

i=1


=4



Æk Ar
i=1 i

1=r

:

ELA

B. Mond and J. Pe
ari

60
For part (b), from (9),



k

Æ As
i=1 i

1s



k

Æ Ar
i=1 i
"

=

"

1=r
k
O

PT

i=1

Remark 3.3. The
ases

P

=

k
O

T

i=1
#1=s

!s

Asi

"
P

k
O

T

i=1

Ari

! #1=r
P

!r #1=r
k
O
 4I:
Ai P

"
PT

P

Ai

! #1=s
P

i=1

k = 2 of the above results were also
onsidered in [7℄.

s = 2 and r = 1, we get
1=2 (M + m)  k 
Æk A2i
Æ Ai
 p
i=1
i=1
2 Mm

3.1. Spe
ial Cases. For


and

For


s = 1 and r =

Æk A2
i=1 i





Æk Ai
i=1

Æk Ai
i=1









M)
 (m4+Mm

Æk Ai 1
i=1

We note that the eigenvalues of

Ai ; i = 1; : : : ; k .

Æk Ai
i=1

2

(M
m)
 4(
I:
M + m)

1, we get


and

1=2

k
O
i=1



Ai

Thus, if the eigenvalues of

1

2



k

Æ A 1
i=1 i



1

p
 ( M pm)2 I:

are the

nk

produ
ts of the eigenvalues of

Ai ; i = 1; : : : ; k , are ordered as

i1  i2  : : :  in > 0; i = 1; : : : ; k;
then the maximum and minimum eigenvalues of

m=

k
Y
i=1

k
O
i=1

in .

Ai

are

This leads to the following four inequalities:

0
1
k
k
Y
Y
B i + i C
nC
1
 k 1=2 B

B
C
k
i
=1
i
=1
2
B
C
Æ Ai
Æ
v

;
A
i
B
C
i=1
k
B u
C i=1
u
Y
 2t i i A
1 n
i=1

M

=

k
Y
i=1

i1

and

ELA

On Inequalities Involving the Hadamard Produ
t of Matri
es



Æk A2
i=1 i

1=2



Æk Ai
i=1

k
Y




4

i=1
k
Y
i=1

(10)



k

Æ Ai
i=1



k
Y



i=1

i1 +

4

k
Y
i=1



Æk Ai







Ai

Æk Ai 1

i=1

1

for

Æk Ai 1
i=1



Ai





1

! I;
in

i=1

Æk Ai 1
i=1

0v
u
k
uY

 t i
i=1

1



1

;

v
12
u
k
u
Y
t i :A I:
n

i=1

in (10), we obtain

k
Y



!2
in

!2
in

i=1
k
Y

i1 +

i1 in

i=1

i=1

Finally, by taking

k
Y

k
Y

i1

61

i=1

i1 +
4

k
Y

i=1

k
Y
i=1

!2
in

i1 in



Æk Ai
i=1



1

:

The inequalities here are generalizations of those given in [4℄. Additional inequalities of a similar kind are possible and will be
onsidered elsewhere.

REFERENCES
[1℄ F.E. Hohn. Elementary Matrix Algebra. Ma
Millan, New York, 1958.
[2℄ T. Kollo and H. Neude
ker. Asymptoti
s of eigenvalues and unit-length eigenve
tors of sample
varian
e and
orrelation matri
es. J. Multivariate Anal., 47:283{300, 1993.
[3℄ S. Liu and H. Neude
ker. Several matrix Kantorovi
h-type inequalities. J. Math. Anal. Appl.,
197:23{26, 1996.
[4℄ B. Mond and J.E. Pe
ari
. Inequalities for the Hadamard produ
t of matri
es. SIAM J. Matrix.
Anal. Appl., 19:66{70, 1998.
[5℄ B. Mond and J.E. Pe
ari
. On Jensen's inequality for operator
onvex fun
tions. Houston J.
Math., 21:739{754, 1995.
[6℄ B. Mond and J.E. Pe
ari
. A matrix version of the Ky Fan inequalities of the Kantorovi
h
inequality II. Linear and Multilinear Algebra, 38:309{313, 1995.
[7℄ Y. Seo, S.E. Takahasi, J. Pe
ari
, and J. Mi
i
. Inequalities of Furuta and Mond-Pe
ari
on the
Hadamard produ
t. J. Inequalities Appl., to appear.