Electronic Journal of Linear Algebra
Volume 32 Volume 32 (2017)
Article 8
2017
Singular Value and Norm Inequalities Associated
with 2 x 2 Positive Semidefinite Block Matrices
Aliaa Burqan
Zarqa University, aliaaburqan@zu.edu.jo
Fuad Kittaneh
University of Jordan, fkitt@ju.edu.jo
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Recommended Citation
Burqan, Aliaa and Kittaneh, Fuad. (2017), "Singular Value and Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block
Matrices", Electronic Journal of Linear Algebra, Volume 32, pp. 116-124.
DOI: https://doi.org/10.13001/1081-3810.3442
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Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 116-124, April 2017.
http://repository.uwyo.edu/ela
SINGULAR VALUE AND NORM INEQUALITIES ASSOCIATED
WITH 2 × 2 POSITIVE SEMIDEFINITE BLOCK MATRICES∗
ALIAA BURQAN† AND FUAD KITTANEH‡
Abstract. This paper aims to give singular value and norm inequalities associated with 2 × 2 positive semidefinite block
matrices.
Key words. Singular value, Positive semidefinite matrix, Block matrix, Unitarily invariant norm, Inequality.
AMS subject classifications. 15A45, 15A60.
1. Introduction. Let Mn (C) denote the space of n × n complex matrices. A norm k.k on Mn (C) is
called unitarily invariant if kU AV k = kAk for all A ∈ Mn (C) and for all unitary matrices U, V ∈ Mn (C).
For Hermitian matrices A, B ∈ Mn (C), we write A ≥ B to mean A − B is positive semidefinite, particularly,
A ≥ 0 indicates that A is positive semidefinite. Likewise, if A is positive definite, we write A > 0. For
A ∈ Mn (C), the singular values of A, denoted by s1 (A) , s2 (A) , . . . , sn (A), are the eigenvalues of the positive
1
semidefinite matrix |A| = (A∗ A) 2 , arranged in decreasing order and repeated according to multiplicity as
s1 (A) ≥ s2 (A) ≥ · · · ≥ sn (A). If A is Hermitian, we label its eigenvalues as λ1 (A) ≥ λ2 (A) ≥ · · · ≥ λn (A).
Several relations between eigenvalues of Hermitian matrices can be obtained by Weyl’s monotonicity principle
[2], which asserts that if A, B are Hermitian and A ≥ B, then
λj (A) ≥ λj (B) for j = 1, . . . , n.
A 0
The direct sum of A and B, denoted by A ⊕ B, is defined to be the block diagonal matrix
.
0 B
Positive semidefinite block matrices play an important role in deriving matrix inequalities. A survey of
results about 2 × 2 positive semidefinite block matrices and related inequalities can be found in Section 7.7
of [8].
A B
It is evident that if A, B, C ∈ Mn (C) are such that
≥ 0, then A and C are positive
B∗ C
semidefinite.
A B
A norm inequality due to Horn and Mathias [7] says that if A, B, C ∈ Mn (C) are such that
≥
B∗ C
0, then for all p > 0 and for every unitarily invariant norm, we have
(1.1)
∗ Received
p 2
k|B| k ≤ kAp k kC p k .
by the editors on November 19, 2016. Accepted for publication on March 17, 2017. Handling Editor: Roger A.
Horn.
† Department of Mathematics, Zarqa University, Zarqa, Jordan (aliaaburqan@zu.edu.jo).
‡ Department of Mathematics, The University of Jordan, Amman, Jordan (fkitt@ju.edu.jo).
Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 116-124, April 2017.
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117
Singular Value and Norm Inequalities
Recently, Audeh and Kittaneh [1] proved that if A, B, C ∈ Mn (C) are such that
A
B∗
B
C
≥ 0, then
sj (B) ≤ sj (A ⊕ C) for j = 1, . . . , n.
(1.2)
Bhatia and Kittaneh [3] proved that if A, B are positive semidefinite, then
(1.3)
1
2
sj A B
1
2
≤ sj
A+B
2
for j = 1, . . . , n.
On the other hand, Hirzallah and Kittaneh have proved in [6] that if A, B ∈ Mn (C), then
A+B
≤ sj (A ⊕ B) for j = 1, . . . , n.
(1.4)
sj
2
In this paper, we are interested in finding singular value versions of the inequality (1.1) . More singular
value inequalities and norm inequalities involving products, sums, and direct sums of matrices based on the
positivity of certain block matrices will be considered.
2. Lemmas. The following lemmas are essential in our analysis. The first lemma is a consequence of the
min-max principle (see [2, p. 75]). The second and third lemmas have been proved in [7]. The fourth lemma
I −I
A B
A+B
0
1
,
follows from the unitary equivalence of
and
. In fact, if U = √2
I I
B A
0
A−B
A B
A+B
0
then U is unitary and
=U
U ∗ (see [5]). The fifth lemma has been proved in
B A
0
A−B
[4]. The sixth lemma is a weak log majorization result that is known as the Weyl majorant theorem (see [2,
p. 42]).
Lemma 2.1. Let A, B, C ∈ Mn (C). Then
sj (ACB) ≤ s1 (A) sj (C) s1 (B) for j = 1, . . . , n.
Lemma 2.2. Let A, B, C ∈ Mn (C) be such that A > 0 and
A
B∗
B
C
≥ 0. Then
B ∗ A−1 B ≤ C.
A B
Lemma 2.3. Let A, B, C ∈ Mn (C) be such that
≥ 0. Then
B∗ C
1
1
B ∗ B ≤ C 2 U ∗ AU C 2 for some unitary matrix U .
Lemma 2.4. Let A, B ∈ Mn (C) be Hermitian. Then
±B ≤ A if and only if
A
B
B
A
≥ 0.
Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 116-124, April 2017.
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Aliaa Burqan and Fuad Kittaneh
118
Lemma 2.5. Let A, B ∈ Mn (C) be Hermitian and such that ±B ≤ A. Then
sj (B) ≤ sj (A ⊕ A) f or j = 1, . . . , n
(2.5)
and
kBk ≤ kAk for every unitarily invariant norm.
(2.6)
Note that the inequality (2.5) is equivalent to saying that
sj (B) ≤ s[ j+1 ] (A) for j = 1, . . . , n,
2
where [k] denotes the integer part of k. In view of Lemma 2.4, this inequality also follows from the inequality
(1.2) .
Lemma 2.6. Let A ∈ Mn (C) have eigenvalues λ1 (A) , λ2 (A) , . . . , λn (A) arranged in such a way that
|λ1 (A)| ≥ |λ2 (A)| ≥ · · · ≥ |λn (A)|. Then
k
Y
|λj (A)| ≤
j=1
k
Y
sj (A) for k = 1, . . . , n − 1,
j=1
with equality for k = n.
3. Main results. In view of the inequality (1.1), one might wonder if the singular value inequality
s2j (B) ≤ sj (A)sj (C) for j = 1, . . . , n
holds true. However, this is refuted by considering A =
1
1
1
2
0
0
#"
2
,C=
1
1
1
1
1
, and B = A 2 C 2 . In this
case,
"
A
1
1
C 2 A2
1
1
A2 C 2
C
#
"
=
1
A2
1
C2
1
A2
1
C2
0
0
#∗
≥0
1 1
and s22 A 2 C 2 ≈ 0.17 > 0.14 ≈ s2 (A)s2 (C).
A B
1
1
≥ 0, then B = A 2 KC 2 for some contraction
B∗ C
matrix K (see [2, p. 269]). Using this and Lemma 2.1, we have
A weaker inequality follows from the fact that if
s2j (B) ≤ min {sj (A)s1 (C), sj (C)s1 (A)} for j = 1, . . . , n.
Our next singular value inequality, which involves a commutativity condition, can be stated as follows.
Electronic Journal of Linear Algebra, ISSN 1081-3810
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Volume 32, pp. 116-124, April 2017.
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119
Singular Value and Norm Inequalities
Theorem 3.1. Let A, B, C ∈ Mn (C) be such that
A
B∗
B
C
≥ 0 and AB = BA. Then
1 1
sj (B) ≤ sj A 2 C 2 for j = 1, . . . , n.
Proof. First assume that A > 0. The general case follows by a continuity argument. Using Lemma 2.2
and the commutativity of A and B, we have
B ∗ B = B ∗ A−1 AB
1
1
= A 2 B ∗ A−1 BA 2
1
1
≤ A 2 CA 2 .
Now, by the Weyl monotonicity principle, we have
1
sj (B) = λj2 (B ∗ B)
1
1
1
≤ λj2 A 2 CA 2
1 1
1
1
1
= λj2 (A 2 C 2 )(A 2 C 2 )∗
1 1
= sj A 2 C 2 for j = 1, . . . , n.
If AB = BA, the singular value inequality in Theorem 3.1 is a refinement of the inequality (1.2). By
the inequalities (1.3) and (1.4), we have
1 1 1
sj (B) ≤ sj A 2 C 2 ≤ sj (A + C) ≤ sj (A ⊕ C) for j = 1, . . . , n.
2
Theorem
3.1#is not true "if the hypothesis
of A and B is omitted. To see this, let A =
"
# of commutativity
√1
√1
0
−
1
A B
1
1
2
2
≥ 0, but s2 (B) ≈ 0.70 > 0.30 ≈ s2 (A 2 C 2 ).
C=
and B =
. Then
∗
√1
√1
1
0
B
C
2
2
A B
≥ 0. Then
Theorem 3.2. Let A, B, C, X, Y ∈ Mn (C) be such that
B∗ C
X ∗ AX + Y ∗ CY ≥ ±(X ∗ BY + Y ∗ B ∗ X).
(3.7)
Proof. Since
X ∗ AX + Y ∗ B ∗ X + X ∗ BY + Y ∗ CY
0
0
0
=
X
Y
0
0
X
−Y
0
0
∗ A
B∗
B
C
A
B∗
B
C
X
Y
0
0
≥0
and
X ∗ AX − Y ∗ B ∗ X − X ∗ BY + Y ∗ CY
0
0
0
=
∗ X
−Y
0
0
≥ 0,
Electronic Journal of Linear Algebra, ISSN 1081-3810
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Aliaa Burqan and Fuad Kittaneh
120
it follows that X ∗ AX + Y ∗ B ∗ X + X ∗ BY + Y ∗ CY ≥ 0 and X ∗ AX − Y ∗ B ∗ X − X ∗ BY + Y ∗ CY ≥ 0.
Thus,
X ∗ AX + Y ∗ CY ≥ ±(X ∗ BY + Y ∗ B ∗ X).
Using Lemma 2.5 and Theorem 3.2, we have the following singular value and norm inequalities involving
sums and direct sums of matrices.
A B
Corollary 3.3. Let A, B, C, X, Y ∈ Mn (C) be such that
≥ 0. Then
B∗ C
(3.8)
sj (X ∗ BY + Y ∗ B ∗ X) ≤ sj ((X ∗ AX + Y ∗ CY ) ⊕ (X ∗ AX + Y ∗ CY )) for j = 1, . . . , n
and
(3.9)
kX ∗ BY + Y ∗ B ∗ Xk ≤ kX ∗ AX + Y ∗ CY k for every unitarily invariant norm.
Letting X = Y = I in Corollary 3.3, we have the following result.
A B
Corollary 3.4. Let A, B, C ∈ Mn (C) be such that
≥ 0. Then
B∗ C
(3.10)
sj (B + B ∗ ) ≤ sj ((A + C) ⊕ (A + C)) for j = 1, . . . , n
and
(3.11)
kB + B ∗ k ≤ kA + Ck for every unitarily invariant norm.
The inequality (3.11) has been recently obtained in [9] using a different argument.
A B
C B∗
A + C B + B∗
If
≥ 0, then
≥ 0, and so
≥ 0. Thus, applying Theorem
B∗ C
B A
B∗ + B A + C
3.1, we have the following result.
A B
Corollary 3.5. Let A, B, C ∈ Mn (C) be such that
≥ 0, AB = BA, and BC = CB. Then
B∗ C
sj (B + B ∗ ) ≤ sj (A + C) for j = 1, . . . , n.
As an application of Corollary 3.5, we have the following singular value inequality for normal matrices.
Corollary 3.6. Let A, B ∈ Mn (C) be normal and such that AB = BA. Then
sj (A∗ B + B ∗ A) ≤ sj (A∗ A + B ∗ B) for j = 1, . . . , n.
∗ A∗ A A∗ B
A 0
A 0
Proof. First observe that
=
≥ 0. Since AB = BA, it follows by
B∗A B∗B
B 0
B 0
∗
the Fuglede-Putnam theorem (see [2, p. 235]) that A B commutes with A∗ A and B ∗ B. Thus, by Corollary
3.5, we have
sj (A∗ B + B ∗ A) ≤ sj (A∗ A + B ∗ B) for j = 1, . . . , n.
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Singular Value and Norm Inequalities
We remark here that
3.6 is not
Corollary
true if the hypothesis of commutativity of A and B is omitted.
1 1
0 1
To see this, let A =
and B =
. Then A and B are normal, but s2 (A∗ B + B ∗ A) =
1 1
−1 0
2 > 1 = s2 (A∗ A + B ∗ B). Moreover, Corollary 3.6 is not true if the hypothesis of normality of A and B is
0 2
1 1
omitted. To see this, let A =
and B =
. Then AB = BA, but s2 (A∗ B + B ∗ A) ≈ 0.83 >
0 0
0 1
0.81 ≈ s2 (A∗ A + B ∗ B).
In spite of the failure of Theorem 3.1 without the hypothesis of commutativity of A and B, using Lemma
2.3 and the Weyl monotonicity principle, one can prove the following related result.
A B
≥ 0. Then
Theorem 3.7. Let A, B, C ∈ Mn (C) be such that
B∗ C
1
1
sj (B) ≤ sj (A 2 U C 2 ) for some unitary matrix U and for j = 1, . . . , n.
Employing Theorem 3.7 and the inequalities (1.3) and (1.4), we can give a different proof of the inequality
(1.2) as follows:
1
1
sj (B) ≤ sj (A 2 U C 2 )
1 1
= sj (A 2 U C 2 )
1
sj (U ∗ AU + C)
2
≤ sj (U ∗ AU ⊕ C)
≤
= sj (A ⊕ C) for j = 1, . . . , n.
4. On the inequality ±B ≤ A. If A, B ∈ Mn (C) are Hermitian and if ±B ≤ A, then the inequality
sj (B) ≤ sj (A) for j = 1, . . . , n,
(4.12)
which is stronger than the inequalities (2.5) and (2.6), need not be true. To see this, let A =
0 1
B=
. Then ±B ≤ A, but s2 (B) = 1 > 0.70 ≈ s2 (A).
1 0
1
1
1
4
and
In spite of the failure of the inequality (4.12), the inequality (2.6) can be strengthened to the following
weak log majorization relation, which can also be concluded from the inequality (2.4) in [7].
Theorem 4.1. Let A, B ∈ Mn (C) be Hermitian and such that ±B ≤ A. Then
k
Y
sj (B) ≤
j=1
Proof. By Lemma 2.4,
A
B
B
A
k
Y
sj (A) for k = 1, . . . , n.
j=1
1
1
≥ 0. Thus, B = A 2 KA 2 for some contraction matrix K, and so for
Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 116-124, April 2017.
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Aliaa Burqan and Fuad Kittaneh
122
k = 1, . . . , n, we have
k
Y
sj (B) =
j=1
k
Y
|λj (B)|
j=1
=
k 1
Y
1
λj A 2 KA 2 j=1
=
k
Y
|λj (AK)|
j=1
≤
k
Y
sj (AK)
(by Lemma 2.6)
j=1
≤
k
Y
sj (A)s1 (K)
(by Lemma 2.1)
j=1
≤
k
Y
sj (A).
j=1
As an application of Theorem 4.1, in view of the inequality (3.7), we have the following result, which is
stronger than the inequality (3.9).
A B
Corollary 4.2. Let A, B, C, X, Y ∈ Mn (C) be such that
≥ 0. Then
B∗ C
k
Y
sj (X ∗ BY + Y ∗ B ∗ X) ≤
k
Y
sj (X ∗ AX + Y ∗ CY ) for k = 1, . . . , n.
j=1
j=1
Letting X = Y = I in Corollary 4.2, we have the following corollary (see [9]), which is stronger than the
inequality (3.11) .
A B
Corollary 4.3. Let A, B, C ∈ Mn (C) be such that
≥ 0. Then
B∗ C
k
Y
sj (B + B ∗ ) ≤
j=1
k
Y
sj (A + C) for k = 1, . . . , n.
j=1
We conclude the paper by observing that the inequalities (2.5) and (3.10) are equivalent.
Theorem 4.4. The following statements are equivalent.
(i) Let A, B ∈ Mn (C) be Hermitian and such that ±B ≤ A. Then
sj (B) ≤ sj (A ⊕ A) for j = 1, . . . , n.
(ii) Let A, B, C ∈ Mn (C) be such that
A
B∗
B
C
≥ 0.Then
sj (B + B ∗ ) ≤ sj ((A + C) ⊕ (A + C)) for j = 1, . . . , n.
Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 116-124, April 2017.
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Singular Value and Norm Inequalities
Proof. (i) =⇒ (ii). Let A, B, C ∈ Mn (C) be such that
A
B∗
B
C
≥ 0. Then it follows by the inequality
(3.7) that ±(B + B ∗ ) ≤ A + C.
Thus, by (i), we have
sj (B + B ∗ ) ≤ sj ((A + C) ⊕ (A + C)) for j = 1, . . . , n.
(ii)=⇒(i). Let A, B ∈ Mn (C) be Hermitian and such that ±B ≤ A. Then, by Lemma 2.4,
A
B
B
A
≥ 0.
Thus, by (ii), we have
sj (2B) ≤ sj (2A ⊕ 2A) ,
and so
sj (B) ≤ sj (A ⊕ A) for j = 1, . . . , n.
In a similar fashion, one can prove the following theorem, which asserts that the inequalities (2.6) and
(3.11) are equivalent.
Theorem 4.5. The following statements are equivalent.
(i) Let A, B ∈ Mn (C) be Hermitian and such that ±B ≤ A. Then
kBk ≤ kAk for every unitarily invariant norm.
(ii) Let A, B, C ∈ Mn (C) be such that
A
B∗
B
C
≥ 0.Then
kB + B ∗ k ≤ kA + Ck for every unitarily invariant norm.
Acknowledgment. The authors are grateful to the referee and Roger Horn for their valuable comments
and suggestions. The work of A. Burqan was funded by the Deanship of Research and Graduate Studies at
Zarqa University, Jordan.
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A publication of the International Linear Algebra Society
Volume 32, pp. 116-124, April 2017.
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Aliaa Burqan and Fuad Kittaneh
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