Journal of Inequalities in Pure and
Applied Mathematics
ON THE STAR PARTIAL ORDERING OF NORMAL MATRICES
JORMA K. MERIKOSKI AND XIAOJI LIU
Department of Mathematics, Statistics and Philosophy
FI-33014 University of Tampere, Finland
EMail: jorma.merikoski@uta.fi
volume 7, issue 1, article 17,
2006.
Received 19 September, 2005;
accepted 09 January, 2006.
Communicated by: S. Puntanen
College of Computer and Information Science
Guangxi University for Nationalities
Nanning 530006, China.
EMail: xiaojiliu72@yahoo.com.cn
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
278-05
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Abstract
We order the space of complex n × n matrices by the star partial ordering ≤∗ .
So A ≤∗ B means that A∗ A = A∗ B and AA∗ = BA∗ . We find several
characterizations for A ≤∗ B in the case of normal matrices. As an application,
we study how A ≤∗ B relates to A2 ≤∗ B2 . The results can be extended to
study how A ≤∗ B relates to Ak ≤∗ Bk , where k ≥ 2 is an integer.
2000 Mathematics Subject Classification: 15A45, 15A18.
Key words: Star partial ordering, Normal matrices, Eigenvalues.
We thank the referee for various suggestions that improved the presentation of this
paper. The second author thanks Guangxi Science Foundation (0575032) for the
support.
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
Title Page
Contents
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Characterizations of A ≤∗ B . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Relationship between A ≤∗ B and A2 ≤∗ B2 . . . . . . . . . . . . 11
4
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References
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1.
Introduction
Throughout this paper, we consider the space of complex n×n matrices (n ≥ 2).
We order it by the star partial ordering ≤∗ . So A ≤∗ B means that A∗ A = A∗ B
and AA∗ = BA∗ . Our motivation rises from the following
Theorem 1.1 (Baksalary and Pukelsheim [1, Theorem 3]). Let A and B be
Hermitian and nonnegative definite. Then A2 ≤∗ B2 if and only if A ≤∗ B.
We cannot drop out the assumption on nonnegative definiteness.
On the Star Partial Ordering of
Normal Matrices
Example 1.1. Let
A=
1 0
,
0 1
B=
1 0
.
0 −1
Jorma K. Merikoski and Xiaoji Liu
Then A2 ≤∗ B2 , but not A ≤∗ B.
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We will study how A ≤∗ B relates to A2 ≤∗ B2 in the case of normal
matrices. We will see (Theorem 3.1) that the “if” part of Theorem 1.1 remains
valid. However, it is not valid for all matrices.
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Example 1.2. Let
A=
1 1
,
0 0
B=
1 1
.
2 −2
Then A ≤∗ B, but not A2 ≤∗ B2 .
In Section 2, we will give several characterizations of A ≤∗ B. Thereafter,
in Section 3, we will apply some of them in discussing our problem. Finally, in
Section 4, we will complete our paper with some remarks.
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2.
Characterizations of A ≤∗ B
Hartwig and Styan ([2, Theorem 2]) presented eleven characterizations of A ≤∗
B for general matrices. One of them uses singular value decompositions. In the
case of normal matrices, spectral decompositions are more accessible.
Theorem 2.1. Let A and B be normal matrices with 1 ≤ rank A < rank B.
Then the following conditions are equivalent:
(a) A ≤∗ B.
(b) There is a unitary matrix U such that
D O
D O
∗
∗
U AU =
,
U BU =
,
O O
O E
where D is a nonsingular diagonal matrix and E 6= O is a diagonal matrix.
(c) There is a unitary matrix U such that
F O
F O
∗
∗
U AU =
,
U BU =
,
O O
O G
where F is a nonsingular square matrix and G 6= O.
(d) If a unitary matrix U satisfies
F O
∗
U AU =
,
O O
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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∗
U BU =
F0 O
,
O G
where F is a nonsingular square matrix, F0 is a square matrix of the same
dimension, and G 6= O, then F = F0 .
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(e) If a unitary matrix U satisfies
D O
∗
U AU =
,
O O
∗
U BU =
D0 O
,
O E
where D is a nonsingular diagonal matrix, D0 is a diagonal matrix of the
same dimension, and E 6= O is a diagonal matrix, then D = D0 .
(f) If a unitary matrix U satisfies
U∗ AU =
D O
,
O O
where D is a nonsingular diagonal matrix, then
D O
∗
U BU =
,
O G
where G 6= O.
(g) All eigenvectors corresponding to nonzero eigenvalues of A are eigenvectors of B corresponding to the same eigenvalues.
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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The reason to assume 1 ≤ rank A < rank B is to omit the trivial cases
A = O and A = B.
Proof. We prove this theorem in four parts.
Part 1. (a) ⇒ (b) ⇒ (c) ⇒ (a).
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(a) ⇒ (b). Assume (a). Then, by normality, A∗ and B commute and are
therefore simultaneously diagonalizable (see, e.g., [3, Theorem 1.3.19]). Since
A and A∗ have the same eigenvectors (see, e.g., [3, Problem 2.5.20]), also A
and B are simultaneously diagonalizable. Hence (recall the assumption on the
ranks) there exists a unitary matrix U such that
0
D O
D O
∗
∗
U AU =
,
U BU =
,
O O
O E
where D is a nonsingular diagonal matrix, D0 is a diagonal matrix of the same
dimension, and E 6= O is a diagonal matrix. Now A∗ A = A∗ B implies
D∗ D = D∗ D0 and further D = D0 . Hence (b) is valid.
(b) ⇒ (c). Trivial.
(c) ⇒ (a). Direct calculation.
Part 2. (a) ⇒ (d) ⇒ (e) ⇒ (a).
This is a trivial modification of Part 1.
Part 3. (b) ⇔ (f).
(b) ⇒ (f). Assume (b). Let U be a unitary matrix satisfying
D O
∗
U AU =
.
O O
By (b), there exists a unitary matrix V such that
0
0
D O
D O
∗
∗
V AV =
,
V BV =
,
O O
O E
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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where D0 is a nonsingular diagonal matrix and E 6= O is a diagonal matrix.
Interchanging the columns
of V if necessary, we can assume D0 = D.
Let U = U1 U2 be such a partition that
∗
∗
U1
U1 AU1 U∗1 AU2
D O
∗
U AU =
A U1 U2 =
=
.
U∗2
U∗2 AU1 U∗2 AU2
O O
Then, for the corresponding partition V = V1 V2 , we have
∗
∗
V1
V1 AV1 V1∗ AV2
D O
∗
V AV =
A V1 V2 =
=
V2∗
V2∗ AV1 V2∗ AV2
O O
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
and
∗
V BV =
∗
V1∗
V1 BV1 V1∗ BV2
D O
B V1 V2 =
=
.
V2∗
V2∗ BV1 V2∗ BV2
O E
Noting that
A = V1
∗
D O
DV∗1
V1
V2
= V1 V2
= V1 DV1∗ ,
O
O O
V2∗
we therefore have
∗
∗
D O
U1
V1
∗
V1 V2
U1 U2
U BU =
∗
∗
U
O E
V2
∗2 ∗
DV1
U1
V1 V2
U1 U2
=
∗
∗
U2
EV2
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DV∗1 U1 DV∗1 U2
=
EV∗2 U1 EV∗2 U2
∗
U1 V1
O
DV∗1 U1
O
=
O
U∗2 V2
O
EV∗2 U2
∗
O
U1 V1 DV∗1 U1
=
O
U∗2 V2 EV∗2 U2
∗
U1 AU1
O
D
O
=
=
,
O
U∗2 V2 EV∗2 U2
O U∗2 V2 EV∗2 U2
U∗1 V1 U∗1 V2
U∗2 V1 U∗2 V2
and so (f) follows.
(f) ⇒ (b). Assume (f). Let U be a unitary matrix such that
D O
∗
U AU =
,
O O
where D is a nonsingular diagonal matrix. Then, by (f),
D O
∗
U BU =
,
O G
where G 6= O. Since G is normal, there exists a unitary matrix W such that
E = W∗ GW is a diagonal matrix. Let
I O
V=U
.
O W
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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Then
I O
I O
∗
V AV =
U AU
O W∗
O W
I O
D O
I O
D O
=
=
O W∗
O O
O W
O O
∗
and
I O
I O
∗
V BV =
U BU
O W∗
O W
I O
D O
I O
D O
=
=
.
O W∗
O G
O W
O E
∗
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
Thus (b) follows.
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Part 4. (b) ⇔ (g).
This is an elementary fact.
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Corollary 2.2. Let A and B be normal matrices. If A ≤∗ B, then AB = BA.
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Proof. Apply (b).
The converse does not hold (even assuming rank A < rank B), see Example 2.1. The normality assumption cannot be dropped out, see Example 2.2.
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Example 2.1. Let
A=
2 0
,
0 0
B=
1 0
.
0 1
J. Ineq. Pure and Appl. Math. 7(1) Art. 17, 2006
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Then AB = BA and rank A < rank B, but A ≤∗ B does not hold. However,
1
A ≤∗ B, which makes us look for a counterexample such that cA ≤∗ B does
2
not hold for any c 6= 0. It is easy to see that we must have n ≥ 3. The matrices




2 0 0
3 0 0
A = 0 3 0 ,
B = 0 4 0
0 0 0
0 0 1
obviously have the required properties.
On the Star Partial Ordering of
Normal Matrices
Example 2.2. Let
A=
0 1
,
0 0
B=
0 1
.
1 0
Jorma K. Merikoski and Xiaoji Liu
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∗
Then A ≤ B, but AB 6= BA.
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3.
Relationship between A ≤∗ B and A2 ≤∗ B2
We will see that A ≤∗ B ⇒ A2 ≤∗ B2 for normal matrices, but the converse
needs an extra condition, which we first present using eigenvalues.
Theorem 3.1. Let A and B be normal matrices with 1 ≤ rank A < rank B.
Then
(a)
A ≤∗ B
On the Star Partial Ordering of
Normal Matrices
is equivalent to the following:
(b)
Jorma K. Merikoski and Xiaoji Liu
A2 ≤∗ B2
and if A and B have nonzero eigenvalues α and respectively β such that α2
and β 2 are eigenvalues of A2 and respectively B2 with a common eigenvector
x, then α = β and x is a common eigenvector of A and B.
Proof. Assuming (a), we have
D O
∗
U AU =
,
O O
∗
U BU =
D O
O E
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as in (b) of Theorem 2.1, and so
2
D O
∗ 2
U A U=
,
O O
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U∗ B2 U =
D2 O
.
O E2
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J. Ineq. Pure and Appl. Math. 7(1) Art. 17, 2006
Hence, by Theorem 2.1, the first part of (b) follows. The second part is trivial.
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Conversely, assume (b). Then
∆ O
∗ 2
U A U=
,
O O
∗
2
U B U=
∆ O
,
O Γ
where U, ∆, and Γ are matrices obtained by applying (b) of Theorem 2.1 to A2
and B2 . Let u1 , . . . , un be the column vectors of U and denote r = rank A.
For i = 1, . . . , r, we have A2 ui = B2 ui = δi ui , where (δi ) = diag ∆. So,
by the second part of (b), there exist complex numbers d1 , . . . , dr such that, for
all i = 1, . . . , r, we have d2i = δi and Aui = Bui = δi ui . Let D be the diagonal
matrix with (di ) = diag D.
For i = r + 1, . . . , n, we have B2 ui = γi−r ui , where (γj ) = diag Γ. Take
complex numbers e1 , . . . , en−r satisfying e2i = γi for i = 1, . . . , n − r. Let E be
the diagonal matrix with (ei ) = diag E. Then
D O
D O
∗
∗
U AU =
,
U BU =
,
O O
O E
and (a) follows from Theorem 2.1.
As an immediate corollary, we obtain a generalization of Theorem 1.1.
Corollary 3.2. Let A and B be normal matrices whose all eigenvalues have
nonnegative real parts. Then A2 ≤∗ B2 if and only if A ≤∗ B.
Next, we present the extra condition using diagonalization.
Theorem 3.3. Let A and B be normal matrices with 1 ≤ rank A < rank B.
Then
(a)
A ≤∗ B
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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is equivalent to the following:
A2 ≤∗ B2
(b)
and if
∗
U AU =
D O
,
O O
∗
U BU =
DH O
,
O E
where U is a unitary matrix, D is a nonsingular diagonal matrix, H is a unitary
diagonal matrix, and E 6= O is a diagonal matrix, then H = I.
(Note that the second part of (b) is weaker than (e) of Theorem 2.1. Otherwise Theorem 3.3 would be nonsense.)
Proof. For (a) ⇒ the first part of (b), see the proof of Theorem 3.1. For (a) ⇒
the second part of (b), see (e) of Theorem 2.1.
Conversely, assume (b). As in the proof of Theorem 3.1, we have
∆ O
∆ O
∗ 2
∗ 2
U A U=
,
U B U=
.
O O
O Γ
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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Hence
U∗ AU =
0
D O
,
O O
U∗ BU =
2
D0 O
,
O E
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0 2
where D and D are diagonal matrices satisfying D = (D ) = ∆ and E is a
diagonal matrix satisfying E2 = Γ.
Denoting (di ) = diag D, (d0i ) = diag D0 , r = rank A, we therefore have
d2i = (d0i )2 for all i = 1, . . . , r. Hence there are complex numbers h1 , . . . , hr
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such that |h1 | = · · · = |hr | = 1 and d0i = di hi for all i = 1, . . . , r. Let H be the
diagonal matrix with (hi ) = diag H. Then D0 = DH, and so D0 = D by the
second part of (b). Thus (b) of Theorem 2.1 is satisfied, and so (a) follows.
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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4.
Remarks
We complete our paper with four remarks.
Remark 1. Let k ≥ 2 be an integer. A natural further question is whether our
discussion can be extended to describe how A ≤∗ B relates to Ak ≤∗ Bk .
As noted by Baksalary and Pukelsheim [1], Theorem 1.1 can be generalized
in a similar way. In other words, for Hermitian nonnegative definite matrices,
Ak ≤∗ Bk if and only if A ≤∗ B. It can be seen also that Theorems 3.1 and 3.3
can be, with minor modifications, extended correspondingly.
Remark 2. Let A and B be arbitrary n × n matrices with rank A < rank B.
Hartwig and Styan ([2, Theorem 2]) proved that A ≤∗ B if and only if there
are unitary matrices U and V such that
Σ O
Σ O
∗
∗
U AV =
,
U BV =
,
O O
O Θ
where Σ is a positive definite diagonal matrix and Θ 6= O is a nonnegative definite diagonal matrix. This is analogous to (a) ⇔ (b) of Theorem 2.1. Actually
it can be seen that all the characterizations of A ≤∗ B listed in Theorem 2.1
have singular value analogies in the general case.
Remark 3. The singular values of a normal matrix are absolute values of its
eigenvalues (see e.g., [3, p. 417]). Hence it is relatively easy to see that if (and
only if) A and B are normal, then U and V above can be chosen so that U∗ V
is a diagonal matrix.
Remark 4. For normal matrices, it can be shown that Theorems 3.1 and 3.3
have singular value analogies. In the proof, it is crucial that U∗ V is a diagonal
matrix. So these results do not remain valid without the normality assumption.
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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References
[1] J.K. BAKSALARY AND F. PUKELSHEIM, On the Löwner, minus and star
partial orderings of nonnegative definite matrices and their squares, Linear
Algebra Appl., 151 (1991), 135–141.
[2] R.E. HARTWIG AND G.P.H. STYAN, On some characterizations on the
“star” partial ordering for matrices and rank subtractivity, Linear Algebra
Appl., 82 (1986), 145–161.
[3] R.A. HORN AND C.R. JOHNSON, Matrix Analysis, Cambridge University
Press, 1985.
On the Star Partial Ordering of
Normal Matrices
Jorma K. Merikoski and Xiaoji Liu
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