Journal of Inequalities in Pure and
Applied Mathematics
MATRIX AND OPERATOR INEQUALITIES
FOZI M. DANNAN
Department of Mathematics
Faculty of Science
Qatar University
Doha - Qatar.
EMail: fmdannan@qu.edu.qa
volume 2, issue 3, article 34,
2001.
Received 3 April, 2001;
accepted 16 May, 2001.
Communicated by: D. Bainov
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
029-01
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Abstract
In this paper we prove certain inequalities involving matrices and operators on
Hilbert spaces. In particular inequalities involving the trace and the determinant
of the product of certain positive definite matrices.
2000 Mathematics Subject Classification: 15A45, 47A50.
Key words: Inequality, Matrix, Operator.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Matrix Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Operator Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References
Matrix and Operator Inequalities
Fozi M. Dannan
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1.
Introduction
Inequalities have proved to be a powerful tool in mathematics , in particular
in modeling error analysis for filtering and estimation problems, in adaptive
stochastic control and for investigation of quantum mechanical Hamiltonians as
it has been shown by Patel and Toda [10, 11, 12] and Lieb and Thirring [5].
It is the object of this paper to prove new interesting matrix and operator
inequalities. We refer the reader to [4, 7, 8] for the basics of matrix and operator
inequalities and for a survey of many other basic and important inequalities.
Through out the paper if A is an n × n matrix, we write trA to denote the
trace of A and det A for the determinant of A. If A is positive definite we write
A > 0. The adjoint of A (a matrix or operator) is denoted by A∗ .
Matrix and Operator Inequalities
Fozi M. Dannan
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2.
Theorem 2.3.
Matrix Inequalities
Through out this section, we work with square matrices on a finite dimensional
Hilbert space.
Theorem 2.1. If A > 0 and B > 0, then
0 < tr (AB)m < (tr (AB))m
(2.1)
for any integer m > 0.
Proof. The equality holds for mP= 1. For m P
> 1, let B = I, and λ1 , λ2 , . . . , λn
m
n
be the eigenvalues of A. Since ni=1 λm
<
(
i
i=1 λi ) , then
0 < tr (Am ) < (trA)m .
(2.2)
Matrix and Operator Inequalities
Fozi M. Dannan
1
1
Since (2.2) is true for any A > 0, we let D = B 2 AB 2 . Then inequality (2.2)
holds for D. Thus 0 < tr (Dm ) < (trD)m , from which the result follows.
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Theorem 2.2. Let A, B be positive definite matrices. Then
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m
s
0 < tr (AB)m < [tr (AB)s ] ,
(2.3)
provided that m and s are positive integers and m > s.
1
1 m
Proof. Clearly tr (AB)m = tr A 2 BA 2
> 0. Let l1 , l2 , . . . , ln be the eigen1
1
1
values of A 2 BA 2 . Then from Hardy’s inequality [3] (l1m + l2m + · · · + lnm ) m <
1
(l1s + l2s + · · · + lns ) s for m > s > 0, we get
h 1
i1
h 1
i1
1 m m
1 s s
tr A 2 BA 2
< tr A 2 BA 2
.
This implies (2.3).
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If Ai > 0 and Bi > 0 (i = 1, 2, . . . , k) , then
tr
(2.4)
k
X
!2
≤
Ai Bi
tr
i=1
k
X
!
A2i
·
tr
k
X
i=1
!
Bi2
.
i=1
If Ai Bi > 0 (i = 1, 2, . . . , k) , then
tr
(2.5)
k
X
!2
Ai Bi
<
tr
i=1
k
X
!
A2i
·
tr
k
X
i=1
!
Bi2
.
Matrix and Operator Inequalities
i=1
Fozi M. Dannan
Proof. Since
0 ≤ tr
k
X
(θAi + Bi )2 = θ2 tr
i=1
k
X
!
A2i +2θtr
k
X
i=1
!
Ai Bi +tr
i=1
k
X
!
Bi2
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,
i=1
we conclude (2.4). To prove (2.5), it suffices to prove that
!2
!2
k
k
X
X
(2.6)
tr
Ai Bi
< tr
(Ai Bi ) .
i=1
i=1
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Pk
Since Ai Bi > 0 for i = 1, 2, . . . , k, then U = i=1 Ai Bi > 0. Therefore the
inequality tr (U )2 < (trU )2 for positive definite U implies (2.6) and the proof
is complete.
Remark 2.1. The condition Ai Bi > 0 in (2.5) is essential as the following
example shows.
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Example 2.1. Let
4 −3
A =
−3 3
3 3
C =
,
3 6
,
2 4
4 9
B=
,
1 −3
B=
.
−3 10
It is clear that A, B, C, and D are positive definite matrices . Now
−10 10
10
560
2
(AB + CD) =
, (AB + CD) =
.
−9 66
−504 4266
Matrix and Operator Inequalities
Fozi M. Dannan
Thus
tr (AB + CD)2 = 4276 > [tr (AB + CD)]2 = 3136.
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Remark 2.2. R. Bellman [1] proved that tr (AB)2 ≤ tr (A2 B 2 ) (*) for positive
definite matrices A and B. Further he asked: “Does the above inequality (*)
hold for higher powers?”. Such a question had been solved by E.H.Lieb and
W.E. Thirring [5] ,where they proved
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(2.7)
tr (AB)m < tr (Am B m )
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for any positive integer m, and for A, B positive definite matrices. In 1995,
Changqin Xu [2] proved a particular case of (2.7): that is when A and B
are 2 × 2 positive definite matrices. Notice that (trAB)m and tr (Am B m )
are upper bounds for tr (AB)m in (2.1) and (2.7) . One may ask what is
max{tr (Am B m ) , tr (AB)m } . The following examples show that either (trAB)m
or tr (Am B m ) can be the least.
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Example 2.2. Let
A=
3 −1
−1 1
,
B=
5 2
2 1
2 1
1 2
.
Then tr (AB)2 = 144 < 204 = tr (A2 B 2 ) .
Example 2.3. Let
A=
3 −2
−2 2
,
B=
.
Matrix and Operator Inequalities
Fozi M. Dannan
Then tr (A2 B 2 ) = 25 < 36 = tr (AB)2 .
Theorem 2.4. If 0 < A1 ≤ B1 and 0 < A2 ≤ B2 , then
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(2.8)
0 < tr (A1 A2 ) ≤ tr (B1 B2 ) .
Proof. Since 0 < A1 ≤ B1 and 0 < A2 ≤ B2 , it follows that
1
1
1
1
0 < A22 A1 A22 ≤ A22 B1 A22
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(2.9)
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1
2
1
2
1
2
1
2
0 < B1 A2 B1 ≤ B1 B2 B1 .
Since trace is a monotone function on the definite matrices, we get
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(2.10)
0 < tr (A1 A2 ) ≤ tr (B1 A2 ) .
and
0 < tr (B1 A2 ) ≤ tr (B1 B2 ) .
(2.11)
This implies (2.8).
Remark 2.3. The conditions A1 > 0 and A2 > 0 in Theorem 2.4. are essential
even if A1 A2 and B1 B2 are symmetric as the following example shows.
Example 2.4. Let
Matrix and Operator Inequalities
−1 1
,
1 −3
1 23
=
,
3
−2
2
A1 =
A2
1 0
B1 =
,
0 2
3 0
B2 =
.
0 2
It is clear that A1 < B1 and A2 < B2 . We have also
1
− 72
3 0
2
A1 A 2 =
,
B1 B2 =
− 72 15
0 4
2
and tr (A1 A2 ) = 8 > 7 = tr (B1 B2 ) .
Theorem 2.5. If A > 0 and B > 0, then
(2.12)
m
n (det A · det B) n ≤ tr (Am B m )
Fozi M. Dannan
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for any positive integer m.
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Proof. Since A is diagonalizable, there exists an orthogonal matrix P and a diagonal matrix Λ such that Λ = P T AP. So if the eigenvalues of A are λ1 , λ2 , . . . , λn ,
then Λ = diag (λ1 , λ2 , . . . , λn ) .
m
Let b11 (m) , b22 (m) , . . . , bnn (m) denote the elements of P BP T . Then
(2.13)
1
1
tr (Am B m ) = tr P Λm P T B m
n
n
1
= tr Λm P T B m P
n
m 1 = tr Λm P T BP
n
1
m
= [λm
b11 (m) + λm
2 b22 (m) + · · · + λn bnn (m)] .
n 1
Using the arithmetic-mean geometric- mean inequality [9], we get
(2.14)
1
1
1
m
m n
n
tr (Am B m ) ≥ [λm
1 λ2 · · · λn ] [b11 (m) b22 (m) · · · bnn (m)] .
n
Since det A ≤ a11 a22 · · · ann for any positive definite matrix A, [4] we conclude
that
m
(2.15)
det P T BP
≤ b11 (m) · b22 (m) · · · · · bnn (m)
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and
(2.16)
Matrix and Operator Inequalities
m
m
det Λm ≤ λm
1 λ2 · · · λn .
J. Ineq. Pure and Appl. Math. 2(3) Art. 34, 2001
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Therefore from (2.14) it follows that
m n1
1
1
tr (Am B m ) ≥ [det (Λm )] n · det P T BP
n
m m
= det P T AP n · det P T BP n
m
= (det A · det B) n .
Here we used the fact that A > 0 and B > 0. The proof is complete.
Corollary 2.6. [6] Let A and X be positive definite n × n- matrices such that
det X = 1. Then
Matrix and Operator Inequalities
Fozi M. Dannan
1
n (det A) n ≤ tr (AX) .
(2.17)
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Proof. Take B = X and m = 1 in Theorem 2.5.
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Theorem 2.7. If A ≥ 0, B ≥ 0 and AB = BA, then
(2.18)
(m−1)n
2
m
m
det (A + B ) ≥ [det (A + B)]
and
(2.19)
m
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2m−1 tr (Am + B m ) ≥ tr (A + B)m
for any positive integer m.
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Proof. To prove inequality (2.18), it is enough to prove
m
Am + B m
A+B
(2.20)
≥
2
2
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for any pair of commuting positive definite matrices A and B. We use induction
to prove (2.20). Clearly (2.20) holds true for m = 2. Assume that (2.20) is true
for m = k. We have to prove (2.20) for m = k + 1. Indeed, since
Ak + B k A + B
A + B Ak + B k
·
=
·
,
2
2
2
2
it follows that
k+1
A+B
A + B Ak + B k
(2.21)
≤
·
2
2
2
Ak+1 + B k+1 Ak+1 + B k+1 BAk + AB k
=
−
+
2
4
4
Ak+1 + B k+1 Ak+1 + B k+1 − BAk − AB k
=
−
2
4
k
k
k+1
k+1
A − B (A − B)
A
+B
=
−
.
2
4
Now the equality
Ak − B k (A − B) = Ak−1 + Ak−2 B + · · · + AB k−2 + B k−1 (A − B)2
for A ≥ 0, B ≥ 0 and AB = BA, implies AB ≥ 0 [8]. Consequently
k−1
k−2
k−1
+B
≥ 0.
Since L · (A − B)2 = (A − B)2 · L, then Ak − B k (A − B) ≥ 0. Therefore,
from (2.21) we obtain
k+1
A+B
Ak+1 + B k+1
≤
.
2
2
L=A
+A
B + · · · + AB
k−2
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Fozi M. Dannan
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The proof is complete. Inequality (2.19) follows directly from (2.20).
Remark 2.4. The condition AB = BA in inequality (2.20) is essential as the
following example shows.
Example 2.5. Let
A=
1 −1
−1 2
,
B=
3 −2
−2 2
.
It is clear that A > 0, B > 0 and AB 6= BA. For m = 3 inequality (2.20)
becomes
(2.22)
4 A3 + B 3 ≥ (A + B)3 .
Easily we find that
3
4 A +B
3
Fozi M. Dannan
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=
Matrix and Operator Inequalities
256 −216
−216 196
,
and det C = −9 which implies that C < 0.
3
(A + B) =
84 −45
−45 24
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3.
Operator Inequalities
In this section we consider inequalities involving operators on separable Hilbert
space X. We start with the following simple well-known inequality.
Theorem 3.1. Let S and T be self-adjoint bounded linear operators on the
Hilbert space H. Then
2
ST + T S
S+T
(3.1)
≤
.
2
2
Proof. Since
2
2
ST + T S
S+T
S+T
−
=
2
2
2
and since the square of the self-adjoint operator is a non-negative operator, we
2
get S+T
≥ 0. The claim of the theorem now follows.
2
Now we present a similar type result as Theorem 3.1 but for non-self-adjoint
case . More precisely:
Theorem 3.2. Let S and T be bounded linear operators on a Hilbert space X.
Assume S to be self-adjoint . Then
1 2
(3.2)
K + Hp ≤ HQ ,
4
where
2
1
S+T
(3.3)
P =
(ST + T S) ,
Q=
,
2
2
1
1
(3.4)
HP =
(P + P ∗ ) ,
HQ = (Q + Q∗ )
2
2
Matrix and Operator Inequalities
Fozi M. Dannan
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and K = 12 (T + T ∗ ) .
Proof. For any bounded linear operator T we have
T =
1
[(T + T ∗ ) + (T − T ∗ )] = HT + K,
2
where HT = 12 (T + T ∗ ) . Inequality (3.1) can be applied to the self-adjoint
operators S and HT , so we get
hU x, xi ≥ 0,
(3.5)
Matrix and Operator Inequalities
Fozi M. Dannan
2
where U = (S + HT ) − 2 (SHT + HT S) . Now we have
(3.6)
U = (S + HT )2 − S (T + T ∗ ) − (T + T ∗ ) S
= (S + HT )2 − 4HP
"
#
2
∗ 2
1
T
+
(T
)
1
=
2S 2 +
+ (T T ∗ + T ∗ T ) − 4HP
2
2
2
1
=
2S 2 + T 2 + (T ∗ )2 − 4HP − 2K 2
2
1
=
8HQ − 4HP − 4HP − 2K 2
2
1 2
= 4 HQ − HP − K .
4
Therefore the required inequality (3.2) follows from (3.6) and (3.5).
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Remark 3.1. When both S and T are not self-adjoint operators , Theorem 3.2
does not hold. The following example illustrates this fact.
Example 3.1. Let S and T be defined on R2 → R2 by the following matrices
1 −2
1 6
S=
,
T =
.
1 4
−3 2
By computation we find that
P =
Q =
HP =
1
HQ − HP − K 2 =
4
1
7 12
(ST + T S) =
,
−6 14
2
81
2 S+T
−1 8
−4
0
2
=
,
K =
,
−4 7
0 − 81
2
4
7 3
−1 2
,
HQ =
,
3 14
2 7
47
− 16 −1
< 0.
−1 − 31
16
Matrix and Operator Inequalities
Fozi M. Dannan
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References
[1] R. BELLMAN, Some inequalities for positive definite matrices, in General Inequalities 2. Proceedings, 2nd Internat. Conf. on General Inequalities”, (E.F.Beckenbach, Ed.), p. 89–90, Birkhauser,1980.
[2] CHNG-QIN XU, Bellman’s inequality, Linear Algebra and its Appl., 229
(1995), 9–14.
[3] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge Univ. Press, Cambridge, 1952.
[4] R.A. HORN
Press, 1999.
AND
C.R. JOHNSON, Matrix Analysis, Cambridge Univ.
[5] E.H. LIEB AND W.E. THIRRING, Inequalities for the moments of the
eigenvalues of the Schrodinger Hamiltonian and their relation to Sobolev
inequalities, Studies in Mathematical Physics, Essays in Honor of Valentine, (1976), Bartmann, Princeton, N.J., 269–303.
[6] J.R. MAGNUS AND H. NEUDECKER, Matrix Differential calculus with
Applications in Statistics and Econometrics, John Wiley & Sons, 1990.
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[7] M. MARCUS AND H. MINC, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Inc., Boston, 1964.
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[8] M.L. MEHTA, Matrix Theory, Selected topics and useful results, Les Editions de Physique, 1989.
Page 16 of 17
[9] D.S. MITRINOVIĆ, Analytic Inequalities, Springer -Verlag Berlin, Heidelberg, New York, 1970 .
J. Ineq. Pure and Appl. Math. 2(3) Art. 34, 2001
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[10] R. PATEL AND M. TODA, Modeling error analysis of stationary linear
discrete time filters, NASA TM X-73, 225(Feb) (1977).
[11] R. PATEL AND M. TODA, Trace inequalities involving Hermitian matrices, Linear Algebra and its Appl., 23 (1979), 13–20.
[12] M. TODA AND R. PATEL, Algorithms for adaptive stochastic control for
a class of linear systems, NASA TM X-73, 240(Apr.) (1977).
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