ON RANK SUBTRACTIVITY BETWEEN NORMAL
MATRICES
JORMA K. MERIKOSKI
XIAOJI LIU
Department of Mathematics and Statistics
FI-33014 University of Tampere,
Finland
EMail: jorma.merikoski@uta.fi
College of Computer and Information Sciences
Guangxi University for Nationalities
Nanning 530006, China
EMail: xiaojiliu72@yahoo.com.cn
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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Contents
Received:
13 July, 2007
Accepted:
05 February, 2008
Communicated by:
JJ
II
F. Zhang
J
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2000 AMS Sub. Class.:
15A45, 15A18.
Page 1 of 18
Key words:
Rank subtractivity, Minus partial ordering, Star partial ordering, Sharp partial
ordering, Normal matrices, EP matrices.
Abstract:
The rank subtractivity partial ordering is defined on Cn×n (n ≥ 2) by A ≤−
B ⇔ rank(B − A) = rank B − rank A, and the star partial ordering by
A ≤∗ B ⇔ A∗ A = A∗ B ∧ AA∗ = BA∗ . If A and B are normal, we
characterize A ≤− B. We also show that then A ≤− B ∧ AB = BA ⇔
A ≤∗ B ⇔ A ≤− B ∧ A2 ≤− B2 . Finally, we remark that some of our
results follow from well-known results on EP matrices.
Acknowledgements:
We thank one referee for alerting us to the results presented in the remark. We
thank also the other referee for his/her suggestions.
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Contents
1
Introduction
3
2
Preliminaries
4
3
Characterizations of A ≤− B
6
4
A ≤− B ∧ AB = BA ⇔ A ≤∗ B
10
5
A ≤− B ∧ A2 ≤− B2 ⇔ A ≤∗ B
12
6
Remarks
16
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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1.
Introduction
The rank subtractivity partial ordering (also called the minus partial ordering) is
defined on Cn×n (n ≥ 2) by
A ≤− B ⇔ rank(B − A) = rank B − rank A.
The star partial ordering is defined by
A ≤∗ B ⇔ A∗ A = A∗ B ∧ AA∗ = BA∗ .
(Actually these partial orderings can also be defined on Cm×n , m 6= n, but square
matrices are enough for us.)
There is a great deal of research about characterizations of ≤∗ and ≤− , see, e.g.,
[8] and its references. Hartwig and Styan [8] applied singular value decompositions to this purpose. In the case of normal matrices, the present authors [10] did
some parallel work and further developments by applying spectral decompostitions
in characterizing ≤∗ . As a sequel to [10], we will now do similar work with ≤− .
In Section 2, we will present two well-known results. The first is a lemma about
a matrix whose rank is equal to the rank of its submatrix. The second is a characterization of ≤− for general matrices from [8].
In Section 3, we will characterize ≤− for normal matrices.
Since ≤∗ implies ≤− , it is natural to ask for an additional condition, which, together with ≤− , is equivalent to ≤∗ . Hartwig and Styan ([8, Theorem 2c]), presented
ten such conditions for general matrices. In Sections 4 and 5, we will find two such
conditions for normal matrices.
Finally, in Section 6, we will remark that some of our results follow from wellknown results on EP matrices.
In [10], we proved characterizations of ≤∗ for normal matrices independently
of general results from [8]. In dealing with the characterization of ≤− for normal
matrices, an independent approach seems too complicated, and so we will apply [8].
Rank Subtractivity Between
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Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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2.
Preliminaries
If 1 ≤ rank A = r < n, then A can be constructed by starting from a nonsingular r × r submatrix according to the following lemma. Since this lemma is of
independent interest, we present it more broadly than we would actually need.
Lemma 2.1. Let A ∈ Cn×n and 1 ≤ r < n, s = n − r. Then the following
conditions are equivalent:
Rank Subtractivity Between
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Jorma K. Merikoski and
(a) rank A = r.
(b) If E ∈ Cr×r is a nonsingular submatrix of A, then there are permutation matrices P, Q ∈ Rn×n and matrices R ∈ Cs×r , S ∈ Cr×s such that
RES RE
A=P
Q.
ES
E
Proof. If (a) holds, then proceeding as Ben-Israel and Greville ([3, p. 178]) gives (b).
Conversely, if (b) holds, then
R
A=P
E S I Q
I
(cf. (22) on [3, p. 178]), and (a) follows.
Next, we recall a characterization of ≤− for general matrices, due to Hartwig and
Styan [8] (and actually stated also for non-square matrices).
Theorem 2.2 ([8, Theorem 1]). Let A, B ∈ Cn×n . If a = rank A, b = rank B,
1 ≤ a < b ≤ n, and p = b − a, then the following conditions are equivalent:
(a) A ≤− B.
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(b) There are unitary matrices U, V ∈ Cn×n such that
Σ O
∗
U AV =
O O
and


Σ + RES RE O
E O ,
U∗ BV =  ES
O
O O
where Σ ∈ Ra×a , E ∈ Rp×p are diagonal matrices with positive diagonal
elements, R ∈ Ca×p , and S ∈ Cp×a .
In fact, U∗ AV is a singular value decomposition of A. (If b = n, then omit the
zero blocks in the representation of U∗ BV.)
Rank Subtractivity Between
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Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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3.
Characterizations of A ≤− B
Now we characterize ≤− for normal matrices.
Theorem 3.1. Let A, B ∈ Cn×n be normal. If a = rank A, b = rank B, 1 ≤ a <
b ≤ n, and p = b − a, then the following conditions are equivalent:
(a) A ≤− B.
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
(b) There is a unitary matrix U ∈ Cn×n such that
D O
∗
U AU =
O O
and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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

D + RES RE O
E O ,
U∗ BU =  ES
O
O O
a×a
where D ∈ C
and S ∈ Cp×a .
,E∈C
p×p
are nonsingular diagonal matrices, R ∈ C
Contents
a×p
,
(c) There is a unitary matrix U ∈ Cn×n such that
G O
∗
U AU =
O O
and


G + RFS RF O
F O ,
U∗ BU =  FS
O
O O
where G ∈ Ca×a , F ∈ Cp×p are nonsingular matrices, R ∈ Ca×p , and S ∈
Cp×a .
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(If b = n, then omit the zero blocks in the representations of U∗ BU.)
Proof. We proceed via (b) ⇒ (c) ⇒ (a) ⇒ (b).
(b) ⇒ (c). Trivial.
(c) ⇒ (a). Assume (c). Then
B − A = UCU∗ ,
where

RFS RF O
F O
C =  FS
O
O O

satisfies
rank C = rank(B − A).
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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On the other hand, by Lemma 2.1,
rank C = rank F = p = b − a = rank B − rank A,
and (a) follows.
(a) ⇒ (b). Assume that A and B satisfy (a). Then, with the notations of Theorem 2.2,
Σ O
∗
= Σ0
U AV =
O O
and


Σ + RES RE O
E O .
U∗ BV =  ES
O
O O
The singular values of a normal matrix are absolute values of its eigenvalues. Therefore the diagonal matrix of (appropriately ordered) eigenvalues of A is D0 = Σ0 J,
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where J is a diagonal matrix of elements with absolute value 1. Furthermore,
V = UJ−1 , and
D O
∗
U AU = D0 =
,
O O
where D is the diagonal matrix of nonzero eigenvalues of A. For details, see, e.g.,
[9, p. 417].
To study U∗ BV, let us denote


K O O
J = O L O  ,
O O M
partitioned as U∗ BV above. Now,



K O O
Σ + RES RE O
E O O L O 
U∗ BU = U∗ BVJ =  ES
O
O O
O O M

 

ΣK + RESK REL O
D + RESK REL O
ESK
EL O =  ESK
EL O .
=
O
O
O
O
O
O
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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By (a),
RESK REL
b − a = rank(B − A) = rank U (B − A)U = rank
.
ESK
EL
∗
Denote E0 = EL. Because E and L are nonsingular, rank E0 = b − a. Hence, by
Lemma 2.1, there are matrices R0 ∈ Ca×p and S0 ∈ Cp×a such that
0 0 0
RESK REL
R E S R0 E 0
=
.
ESK
EL
E 0 S0
E0
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Consequently,


D + R 0 E 0 S0 R 0 E 0 O
E 0 S0
E0 O ,
U∗ BU = 
O
O O
and (b) follows.
Corollary 3.2. Let A, B ∈ Cn×n . If A is normal, B is Hermitian, and A ≤− B,
then A is Hermitian.
Proof. If rank A = 0 or rank A = rank B, the claim is trivial. Otherwise, with the
notations of Theorem 3.1,


D + RES RE O
D O
E O .
A0 = U∗ AU =
,
B0 = U∗ BU =  ES
O O
O
O O
Since B is Hermitian, B0 is also Hermitian. Therefore E∗ = E and ES = (RE)∗ =
ER∗ , which implies S = R∗ , since E is nonsingular. Now


RER∗ RE O
E O
A0 = B0 −  ER∗
O
O O
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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is a difference of Hermitian matrices and so Hermitian. Hence also A is Hermitian.
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4.
A ≤− B ∧ AB = BA ⇔ A ≤∗ B
The partial ordering ≤∗ implies ≤− . For the proof, apply Theorem 2.2 and the
corresponding characterization of ≤∗ ([8, Theorem 2]). In fact, this implication
originates with Hartwig ([7, p. 4, (iii)]) on general star-semigoups.
We are therefore motivated to look for an additional condition, which, together
with ≤− , is equivalent to ≤∗ . First we recall a characterization of ≤∗ from [10] but
formulate it slightly differently.
Theorem 4.1 ([10, Theorem 2.1ab], cf. also [8, Theorem 2ab]). Let A, B ∈ Cn×n
be normal. If a = rank A, b = rank B, 1 ≤ a < b ≤ n, and p = b − a, then the
following conditions are equivalent:
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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(a) A ≤∗ B.
(b) There is a unitary matrix U ∈ Cn×n such that
D O
∗
U AU =
O O
and
Rank Subtractivity Between
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Jorma K. Merikoski and
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Page 10 of 18


D O O
U∗ BU = O E O ,
O O O
where D ∈ Ca×a and E ∈ Cp×p are nonsingular diagonal matrices. (If b = n,
then omit the third block-row and block-column of zeros in the expression of
B.)
Hartwig and Styan [8] proved the following theorem assuming that A and B are
Hermitian. We assume only normality.
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Theorem 4.2 (cf. [8, Corollary 1ac]). Let A, B ∈ Cn×n be normal. The following
conditions are equivalent:
(a) A ≤∗ B,
(b) A ≤− B ∧ AB = BA.
Proof. If a = rank A and b = rank B satisfy a = 0 or a = b, then the claim is
trivial. So we assume 1 ≤ a < b ≤ n.
(a) ⇒ (b). This follows immediately from Theorems 4.1 and 3.1.
(b) ⇒ (a). Assume (b). Since A ≤− B, we have with the notations of Theorem 3.1




D + RES RE O
D O O
E O .
U∗ AU = O O O ,
U∗ BU =  ES
O O O
O
O O
 2

Thus
D + DRES DRE O
O
O
O
U∗ ABU = 
O
O
O


and
D2 + RESD O O
∗
ESD
O O .
U BAU = 
O
O O
Since AB = BA, also U∗ ABU = U∗ BAU, which implies DRE = O and
ESD = O. Because D and E are nonsingular, we therefore have R = O and


S = O. So
D O O
U∗ BU = O E O ,
O O O
and (a) follows from Theorem 4.1.
Rank Subtractivity Between
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Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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5.
A ≤− B ∧ A2 ≤− B2 ⇔ A ≤∗ B
We first note that the conditions A ≤− B and A2 ≤− B2 are independent, even if
A and B are Hermitian.
Example 5.1. If
A=
1 0
,
0 0
B=
5 2
,
2 1
then
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
Xiaoji Liu
4 2
rank(B − A) = rank
= 1,
2 1
vol. 9, iss. 1, art. 4, 2008
rank B − rank A = 2 − 1 = 1,
and so A ≤− B. However, A2 ≤− B2 does not hold, since
1 0
29 12
28 12
2
2
2
2
A =
,
B =
,
B −A =
,
0 0
12 5
12 5
rank B2 − A2 = 2,
rank B2 − rank A2 = 2 − 1 = 1.
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Page 12 of 18
Example 5.2. If
A=
1 0
,
0 0
B=
−1 0
,
0 0
then A2 ≤− B2 holds but A ≤− B does not hold.
Gross ([5, Theorem 5]) proved that, in the case of Hermitian nonnegative definite
matrices, the conditions A ≤− B and A2 ≤− B2 together are equivalent to A ≤∗ B.
Baksalary and Hauke ([1, Theorem 4]) proved it for all Hermitian matrices. We
generalize this result.
Theorem 5.1. Let A, B ∈ Cn×n be normal. Assume that
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(i) B is Hermitian
or
(ii) B − A is Hermitian.
Then the following conditions are equivalent:
(a) A ≤∗ B,
(b) A ≤− B ∧ A2 ≤− B2 .
Proof. First, assume (i). If A ≤− B, then A is Hermitian by Corollary 3.2. If
A ≤∗ B, then A ≤− B, and so A is Hermitian also in this case. Therefore, both
(a) and (b) imply that A is actually Hermitian, and hence (a) ⇔ (b) follows from [1,
Theorem 4]. The following proof applies to an alternative.
Second, assume (ii). If a = rank A and b = rank B satisfy a = 0 or a = b, then
the claim is trivial. So we let 1 ≤ a < b ≤ n.
(a) ⇒ (b). This is an immediate consequence of Theorems 4.1 and 3.1.
(b) ⇒ (a). Assume (b). Since A ≤− B, we have with the notations of Theorem 3.1


D + RES RE O
D O
E O U∗ .
A=U
U∗ ,
B = U  ES
O O
O
O O
Since B−A is Hermitian, U∗ (B−A)U is also Hermitian. Therefore E is Hermitian
and S = R∗ , and so


D + RER∗ RE O
E O U∗ .
B = U  ER∗
O
O O
Rank Subtractivity Between
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Jorma K. Merikoski and
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vol. 9, iss. 1, art. 4, 2008
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Furthermore,
D2 O
U∗
A =U
O O
2
and

(D + RER∗ )2 + RE2 R∗ (D + RER∗ )RE + RE2 O
B2 = U ER∗ (D + RER∗ ) + E2 R∗
ER∗ RE + E2
O U∗ .
O
O
O

Now
Rank Subtractivity Between
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Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
where
DRER∗ + RER∗ D + (RER∗ )2 + RE2 R∗ DRE + RER∗ RE + RE2
H=
.
ER∗ D + ER∗ RER∗ + E2 R∗
ER∗ RE + E2
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H O
2
2
B −A =U
U∗ ,
O O
Multiplying the second block-row of H by −R from the right and adding the result
to the first block-row is a set of elementary row operations and so does not change
the rank. Thus
DRER∗
DRE
rank H = rank
= rank H0 .
ER∗ D + ER∗ RER∗ + E2 R∗ ER∗ RE + E2
Furthermore, multiplying the second block-column of H0 by −R∗ from the right and
adding the result to the first block-column is a set of elementary column operations,
and so
O
DRE
0
rank H = rank
= rank H00 .
ER∗ D ER∗ RE + E2
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Since A2 ≤− B2 , we therefore have
rank H00 = rank(B2 − A2 ) = rank B2 − rank A2 = b − a = p.
Because ER∗ RE is Hermitian nonnegative definite and E is Hermitian positive definite, their sum E0 = ER∗ RE + E2 is Hermitian positive definite and hence nonsingular. Applying Lemma 2.1 to H00 , we see that there is a matrix S ∈ Cp×a such that
(1) S∗ E0 = DRE and (2) S∗ E0 S = O. Since E0 is positive definite, then (2) implies
S = O, and so (1) reduces to DRE = O, which, in turn, implies R = O by the
nonsingularity of D and E. Consequently,


D O O
B = U O E O U∗ ,
O O O
Rank Subtractivity Between
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Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
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and (a) follows from Theorem 4.1.
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6.
Remarks
A matrix A ∈ Cn×n is a group matrix if it belongs to a subset of Cn×n which is a
group under matrix multiplication. This happens if and only if rank A2 = rank A
(see, e.g., [3, Theorem 4.2] or [11, Theorem 9.4.2]). A matrix A ∈ Cn×n is an EP
matrix if R(A∗ ) = R(A) where R denotes the column space. There are plenty
of characterizations for EP matrices, see Cheng and Tian [4] and its references. A
normal matrix is EP, and an EP matrix is a group matrix (see, e.g., [3, p. 159]). The
sharp partial ordering between group matrices A and B is defined by
A ≤# B ⇔ A2 = AB = BA.
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
Xiaoji Liu
vol. 9, iss. 1, art. 4, 2008
Three of our results follow from well-known results on EP matrices.
First, Corollary 3.2 is a special case of Lemma 3.1 of Baksalary et al [2], where
A is assumed only EP.
Second, let A and B be group matrices. Then
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A ≤# B ⇔ A ≤− B ∧ AB = BA,
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by Mitra ([12, Theorem 2.5]). On the other hand, if A is EP, then
A ≤# B ⇔ A ≤∗ B,
by Gross ([6, Remark 1]). Hence Theorem 4.2 follows assuming only that A is EP
and B is a group matrix.
Third, Theorem 5.1 with assumption (i) is a special case of [2, Corollary 3.2],
where A is assumed only EP.
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References
[1] J.K. BAKSALARY AND J. HAUKE, Characterizations of minus and star orders
between the squares of Hermitian matrices, Linear Algebra Appl., 388 (2004),
53–59.
[2] J.K. BAKSALARY, J. HAUKE, X. LIU AND S. LIU, Relationships between
partial orders of matrices and their powers, Linear Algebra Appl., 379 (2004),
277–287.
Rank Subtractivity Between
Normal Matrices
Jorma K. Merikoski and
Xiaoji Liu
[3] A. BEN-ISRAEL AND T.N.E. GREVILLE, Generalized Inverses. Theory and
Applications, Second Edition. Springer, 2003.
vol. 9, iss. 1, art. 4, 2008
[4] S. CHENG AND Y. TIAN, Two sets of new characterizations for normal and
EP matrices, Linear Algebra Appl., 375 (2003), 181–195.
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[5] J. GROSS, Löwner partial ordering and space preordering of Hermitian nonnegative definite matrices, Linear Algebra Appl., 326 (2001), 215–223.
[6] J. GROSS, Remarks on the sharp partial order and the ordering of squares of
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[7] R.E. HARTWIG, How to partially order regular elements, Math. Japonica, 25
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[9] R.A. HORN
Press, 1985.
AND
C.R. JOHNSON, Matrix Analysis, Cambridge University
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[10] J.K. MERIKOSKI AND X. LIU, On the star partial ordering of normal matrices,
J. Ineq. Pure Appl. Math., 7(1) (2006), Art. 17. [ONLINE: http://jipam.
vu.edu.au/article.php?sid=647].
[11] L. MIRSKY, An Introduction to Linear Algebra, Clarendon Press, 1955.
Reprinted by Dover Publications, 1990.
[12] S.K. MITRA, On group inverses and their sharp order, Linear Algebra Appl., 92
(1987), 17–37.
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vol. 9, iss. 1, art. 4, 2008
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