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 OFMATRICES
B. MOND
y
AND J. PE CARI
C z
Abstrat. Reently,theauthorsestablishedanumberofinequalitiesinvolvingintegerpowersof theHadamardprodutoftwopositivedeniteHermitianmatries. Heretheseresultsareextended intwo ways. First, the restritiontointeger powersisrelaxedtoinlude allrealnumbersnotin theopeninterval( 1;1). Seond,theresultsareextended totheHadamardprodutofanynite numberofHermitianpositivedenitematries.
Keywords.matrixinequalities,Hadamardprodut AMSsubjet lassiations.15A45
1. Introdution. LetAandB bennmatries. AÆB denotestheHadamard produtandAB theKronekerprodutofA andB.
Thesetwoprodutsarerelatedbythefollowingrelation[2℄,[3℄. Thereexistsann
2
nseletionmatrixJ suhthatJ T
J =I and AÆB=J
T
(AB)J: NotethatJ
T
isthenn 2
matrix[E 11
E 22
:::E nn
℄, whereE ii
isthennmatrixof zerosexeptforaonein the(i;i)thposition.
Using this result, in [4℄ the authors proved a number of inequalities involving integerpowersoftheHadamardprodutoftwopositivedeniteHermitianmatries. Here we extend these results in two ways. First, the restrition to integer powers is relaxed to inlude all real numbersnot in the open interval ( 1;1). Seond, the resultsareextendedtotheHadamardprodutofanynitenumberofnnHermitian positivedenitematries.
2. NotationandPreliminaryResults. TheHadamardandKroneker prod-utsofmatriesA
i
; i=1;:::k,willbedenoted by k Æ i=1
and k O i=1
,respetively. WeshallmakefrequentuseofthefollowingpropertyoftheKronekerprodut:
(AB)(CD)=(AC)(BD): ForanitenumberofmatriesA
i ; B
i
; i=1;:::;k,thisbeomes k
Y i=1
A i
!
k Y i=1
B i
! =
k Y i=1
(A i
B i
): (1)
Reeivedby the editors on 11 Otober1999. Aepted for publiation on 25January 2000. Handlingeditor:DanielHershkowitz.
y
Department of Mathematis, La Trobe University, Bundoora, Vitoria 3083, Australia (matbmlura.latrobe.edu.au).
z
ELA
OnInequalitiesInvolvingtheHadamardProdutofMatries 57 LetA beapositivedenite nn Hermitianmatrix. Thereexists amatrix U suh that
A=U
[ 1
; 2
;:::; n
℄U; U
U =I; where [
1 ;
2 ;:::;
n
℄ is the diagonalmatrix with i
, the positiveeigenvaluesof A, alongthediagonal[1℄. For anyrealnumbers; A
s
isdenedby A
s =U
[
s 1
; s 2
;:::; s n
℄U:
Lemma 2.1. LetAandB bepositive deniteHermitiannn matries andsa nonzerorealnumber. Then
A s
B s
=(AB) s
: (2)
Proof. Assume
B=V
[ 1
; 2
;:::; n
℄V; V
V =I; where
i
aretheeigenvaluesofB. Then A
s B
s =(U
[
s 1
;:::; s n
℄U)(V
[ s 1
;:::; s n
℄V) =(U
V
)([
s 1
;:::; s n
℄[ s 1
;:::; s n
℄)(UV) =(UV)
([
s 1
;:::; s n
℄[ s 1
;:::; s n
℄)(UV)=(AB) s
: Notethat
(UV)
(UV) =(U
V
)(UV) =(U
U)(V
V)=II =I n
2:
Equation(2)extendsreadily,foranitenumberofnnpositivedeniteHermitian matriesA
i
; i=1;:::;k,to k O i=1
(A s i
)= k O
i=1 A
i !
s : (3)
Lemma 2.2. Let A i
; i = 1;:::;k, be nn matries. There exists an n k
n seletion matrixP suhthatP
T P =I and
k Æ i=1
A i
=P T
k O i=1
A i
! P: (4)
Weprovethisforthreematries. 1
Theextensionfrom mtom+1issimilar. AÆBÆC =AÆJ
T
(BC)J =J
T
(A(J T
(BC)J))J =J
T
((IAI)(J T
(BC)J))J =J
T (IJ
T
)(ABC)(IJ)J by(1) =J
T (IJ)
T
(ABC)(IJ)J =
^ J
T
ELA
58 B.MondandJ.Peari where
^
J isthen 3
nmatrix ^
J =(IJ)J. Notethat ^ J
T ^ J =I. 3. Results. Inthissetion,A
i
; i=1;:::;k,willdenote nnpositivedenite Hermitianmatries. A
i A
j
meansthatA i
A j
ispositivesemidenite.
Theorem 3.1. Let r and s bereal numbers r<s,and either r2=( 1;1) and s2=( 1;1)or s1r
1 2 orr 1s
1 2
. Then
k Æ i=1
A s i
1=s
k Æ i=1
A r i
1=r : (5)
Proof. Wemakeuseofthefollowingresult[5℄.
LetAbeannnpositivedeniteHermitianmatrixandletV beanntmatrix suhthatV
V =I. Then (V
A
s V)
1=s (V
A
r V)
1=r
forallrealrands,r<s,suhthateitherr2=( 1;1)ands2=( 1;1)ors1r 1 2 orr 1s
1 2 .
HereinsteadofV,weusethen k
nseletionmatrixP givenby(4). Noting(3), wehave
k Æ i=1
A s i
1=s =
P
t
N k i=1
A s i
P
1=s =
P
t
N k i=1
A i
s
P
1=s
P
T
N k i=1
A i
r
P
1=r =
P
t
N k i=1
A r i
P
1=r =
k Æ i=1
A r i
1=r : Somespeial asesof(5)arethefollowing:
k Æ i=1
A 1 i
1
k Æ i=1
A i
or,equivalently
k Æ i=1
A i
1
k Æ i=1
A 1 i
: Forr>1,wehave
k Æ i=1
A i
k Æ i=1
A r i
1=r
or,equivalently,
k Æ i=1
A 1=r i
k Æ i=1
A i
1=r
: Forr=2,thelast twoinequalitiesbeome
k Æ
A i
k Æ
A 2 i
ELA
OnInequalitiesInvolvingtheHadamardProdutofMatries 59 and
k Æ i=1
A 1=2 i
k Æ i=1
A i
1=2
:
Theorem 3.2. Let r and s be nonzero real numbers suh that s > r and s2=( 1;1)or r2=( 1;1). Then
(a)
k Æ i=1
A s i
1=s
4
k Æ i=1
A r i
1=r ; where
4=
r(
s
r ) (s r)(
r 1)
1=s
s(
r
s ) (r s)(
s 1)
1=r
; (6)
=M=m, and M and m are, respetively, the largest and smallest eigenvalues of k
O i=1
A i
. Also,
(b)
k Æ i=1
A s i
1=s
k Æ i=1
A r i
1=r 4I; where
4= max 2[0;1℄
f[M s
+(1 )m s
℄ 1=s
[M r
+(1 )m r
℄ 1=r
g: (7)
Proof. Let A be an nn positive denite Hermitian matrix with eigenvalues ontainedin theinterval [m;M℄, where 0<m<M, and let V bean nt matrix suhthat V
V =I. Ifr ands arenonzerorealnumberssuh that r<sandeither s2=( 1;1)orr2=( 1;1), then[6℄
(V
A s
V) 1=s
4(V
A
r V)
1=r (8)
where
4isgivenby(6), and (V
A
s V)
1=s (V
A
r V)
1=r 4I (9)
where4isgivenby(7). Thusforpart(a),from(8)andnoting(3)and(4),wehave
k Æ i=1
A s i
1=s =
" P
T k O
i=1 A
s i !
P #
1=s =
" P
T k O i=1
A i
! s
P #1=s
4
" P
T k O
A i
! r
P #1=r
= 4
" P
T k O
A r i
! P
# 1=r
= 4
k Æ i=1
A r i
ELA
Remark 3.3. Theasesk=2oftheaboveresultswerealsoonsideredin [7℄. 3.1. Speial Cases. Fors=2andr=1,weget
Wenotethat theeigenvaluesof k
produtsoftheeigenvaluesof A
thenthemaximumandminimumeigenvaluesof k
. Thisleadstothefollowingfourinequalities:
ELA
OnInequalitiesInvolvingtheHadamardProdutofMatries 61
k Æ i=1
A 2 i
1=2
k Æ i=1
A i
k Y i=1
i 1
k Y i=1
i n
! 2
4 k Y i=1
i 1
+ k Y i=1
i n
! I;
k Æ i=1
A i
k Y i=1
i 1
+ k Y i=1
i n
!2
4 k Y i=1
i 1
i n
k Æ i=1
A 1 i
1
; (10)
k Æ i=1
A i
k Æ i=1
A 1 i
1
0
v u u t
k Y i=1
i 1
v u u t k Y i=1
i n
: 1 A 2
I:
Finally,bytakingA 1 i
forA i
in(10),weobtain
k Æ i=1
A 1 i
k Y i=1
i 1
+ k Y i=1
i n
! 2
4 k Y i=1
i 1
i n
k Æ i=1
A i
1
:
Theinequalitiesherearegeneralizationsofthosegivenin[4℄. Additional inequal-itiesofasimilarkindarepossibleandwillbeonsideredelsewhere.
REFERENCES
[1℄ F.E.Hohn. ElementaryMatrixAlgebra.MaMillan,NewYork,1958.
[2℄ T.KolloandH.Neudeker. Asymptotisofeigenvaluesandunit-lengtheigenvetorsofsample varianeandorrelationmatries. J.MultivariateAnal.,47:283{300,1993.
[3℄ S.LiuandH.Neudeker. SeveralmatrixKantorovih-typeinequalities. J.Math.Anal.Appl., 197:23{26,1996.
[4℄ B.MondandJ.E.Peari.InequalitiesfortheHadamardprodutofmatries.SIAMJ.Matrix. Anal.Appl.,19:66{70,1998.
[5℄ B.MondandJ.E.Peari. OnJensen'sinequalityforoperatoronvex funtions. Houston J. Math.,21:739{754,1995.
[6℄ B.Mond and J.E. Peari. Amatrix version ofthe KyFan inequalities of the Kantorovih inequalityII. LinearandMultilinearAlgebra,38:309{313,1995.
