Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 1, Article 17, 2006
ON THE STAR PARTIAL ORDERING OF NORMAL MATRICES
JORMA K. MERIKOSKI AND XIAOJI LIU
D EPARTMENT OF M ATHEMATICS , S TATISTICS AND P HILOSOPHY
FI-33014 U NIVERSITY OF TAMPERE , F INLAND
jorma.merikoski@uta.fi
C OLLEGE OF C OMPUTER AND I NFORMATION S CIENCE
G UANGXI U NIVERSITY FOR NATIONALITIES
NANNING 530006, C HINA
xiaojiliu72@yahoo.com.cn
Received 19 September, 2005; accepted 09 January, 2006
Communicated by S. Puntanen
A BSTRACT. 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.
Key words and phrases: Star partial ordering, Normal matrices, Eigenvalues.
2000 Mathematics Subject Classification. 15A45, 15A18.
1. I NTRODUCTION
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.
Example 1.1. Let
A=
1 0
,
0 1
B=
1 0
.
0 −1
Then A2 ≤∗ B2 , but not A ≤∗ B.
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
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.
278-05
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J ORMA K. M ERIKOSKI AND X IAOJI L IU
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.
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.
2. C HARACTERIZATIONS 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
0
F O
F O
∗
∗
U AU =
,
U BU =
,
O 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 .
(e) If a unitary matrix U satisfies
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, then D = D0 .
(f) If a unitary matrix U satisfies
D O
∗
U AU =
,
O O
where D is a nonsingular diagonal matrix, then
D O
∗
U BU =
,
O G
where G 6= O.
J. Inequal. Pure and Appl. Math., 7(1) Art. 17, 2006
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S TAR PARTIAL O RDERING
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3
(g) All eigenvectors corresponding to nonzero eigenvalues of A are eigenvectors of B corresponding to the same eigenvalues.
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).
(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
where D0 is a nonsingular diagonal matrix and E 6= O is a diagonal matrix.
columns of V if necessary,
we can assume D0 = D.
Let U = U1 U2 be such a partition that
∗
∗
D
U1 AU1 U∗1 AU2
U1
∗
=
A U1 U2 =
U AU =
O
U∗2 AU1 U∗2 AU2
U∗2
Then, for the corresponding partition V = V1 V2 , we have
∗
∗
D
V1 AV1 V1∗ AV2
V1
∗
=
A V1 V2 =
V AV =
O
V2∗ AV1 V2∗ AV2
V2∗
Interchanging the
O
.
O
O
O
and
V BV =
∗
D O
V1 BV1 V1∗ BV2
V1∗
=
.
B V1 V2 =
O E
V2∗ BV1 V2∗ BV2
V2∗
A = V1
∗
D O
DV∗1
V1
V2
= V1 V2
= V1 DV1∗ ,
O O
V2∗
O
∗
Noting that
J. Inequal. Pure and Appl. Math., 7(1) Art. 17, 2006
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J ORMA K. M ERIKOSKI AND X IAOJI L IU
we therefore have
U∗1
U∗2
∗
D O
V1
V1 V2
U1 U2
U BU =
∗
O E
V2
∗
∗
DV1
U1
V1 V2
U1 U2
=
∗
∗
U2
EV2
∗
U1 V1 U∗1 V2
DV∗1 U1 DV∗1 U2
=
U∗2 V1 U∗2 V2
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
∗
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
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
∗
U BU
V BV =
O W
O W∗
D O
I O
D O
I O
=
.
=
O G
O W
O E
O W∗
∗
Thus (b) follows.
Part 4. (b) ⇔ (g).
This is an elementary fact.
Corollary 2.2. Let A and B be normal matrices. If A ≤∗ B, then AB = BA.
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.
J. Inequal. Pure and Appl. Math., 7(1) Art. 17, 2006
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S TAR PARTIAL O RDERING
OF
N ORMAL M ATRICES
5
Example 2.1. Let
A=
2 0
,
0 0
B=
1 0
.
0 1
Then AB = BA and rank A < rank B, but A ≤∗ B does not hold. However, 21 A ≤∗ B,
which makes us look for a counterexample such that cA ≤∗ B does 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.
Example 2.2. Let
A=
0 1
,
0 0
B=
0 1
.
1 0
Then A ≤∗ B, but AB 6= BA.
3. R ELATIONSHIP 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 ≤∗ B
(a)
is equivalent to the following:
A2 ≤∗ B2
(b)
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
∗
U AU =
D O
,
O O
∗
U BU =
D O
O E
as in (b) of Theorem 2.1, and so
∗
2
U A U=
D2 O
,
O O
∗
2
U B U=
D2 O
.
O E2
Hence, by Theorem 2.1, the first part of (b) follows. The second part is trivial.
Conversely, assume (b). Then
∆ O
∆ O
∗ 2
∗ 2
U A U=
,
U B U=
,
O 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.
J. Inequal. Pure and Appl. Math., 7(1) Art. 17, 2006
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J ORMA K. M ERIKOSKI AND X IAOJI L IU
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 E
O O
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 ≤∗ B
(a)
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
Hence
∗
U AU =
D O
,
O O
∗
U BU =
D0 O
,
O E
where D and D0 are diagonal matrices satisfying D2 = (D0 )2 = ∆ 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 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.
J. Inequal. Pure and Appl. Math., 7(1) Art. 17, 2006
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S TAR PARTIAL O RDERING
OF
N ORMAL M ATRICES
7
4. R EMARKS
We complete our paper with four remarks.
Remark 4.1. 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 4.2. 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 4.3. 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 4.4. 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.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 7(1) Art. 17, 2006
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