vol13_pp162-174. 208KB Jun 04 2011 12:05:53 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
SUBDIRECT SUMS OF NONSINGULAR
M -MATRICES
AND OF
THEIR INVERSES∗
RAFAEL BRU† , FRANCISCO PEDROCHE† ,
AND
DANIEL B. SZYLD‡
Abstra
t. The question of when the subdire
t sum of two nonsingular M -matri
es is a nonsingular M -matrix is studied. Su
ient
onditions are given. The
ase of inverses of M -matri
es is
also studied. In parti
ular, it is shown that the subdire
t sum of overlapping prin
ipal submatri
es
of a nonsingular M -matrix is a nonsingular M -matrix. Some examples illustrating the
onditions
presented are also given.
AMS subje
t
lassi
ations. 15A48.
Key words. Subdire
t sum, M -matri
es, Inverse of M -matrix, Overlapping blo
ks.
1. Introdu
tion. Subdire
t sum of matri
es are generalizations of the usual sum
of matri
es (a k-subdire
t sum is formally de ned below in Se
tion 2). They were
introdu
ed by Fallat and Johnson in [3℄, where many of their properties were analyzed.
For example, they showed that the subdire
t sum of positive de nite matri
es, or of
symmetri
M -matri
es, are positive de nite or symmetri
M -matri
es, respe
tively.
They also showed that this is not the
ase for M -matri
es: the sum of two M -matri
es
may not be an M -matrix. One goal of the present paper is to give su
ient
onditions
so that the subdire
t sum of nonsingular M -matri
es is a nonsingular M -matrix. We
also treat the
ase of the subdire
t sum of inverses of M -matri
es.
Subdire
t sums of two overlapping prin
ipal submatri
es of a nonsingular M matrix appear naturally when analyzing additive S
hwarz methods for Markov
hains
or other matri
es [2℄, [4℄. In this paper we show that the subdire
t sum of two
overlapping prin
ipal submatri
es of a nonsingular M -matrix is a nonsingular M matrix.
The paper is stru
tured as follows. In Se
tion 2 we fo
us on the nonsingularity of
the subdire
t sum of any pair of nonsingular matri
es, giving an expli
it expression
for the inverse. In Se
tion 2.1 we study the k-subdire
t sum of two nonsingular
M -matri
es and in parti
ular, the
ase of subdire
t sums of overlapping blo
ks of
nonsingular M -matri
es. In Se
tion 2.3 we extend some results to the subdire
t sum
of more than two nonsingular M -matri
es. In Se
tion 3 we analyze the subdire
t sum
of two inverses. Finally, in Se
tion 4 we mention some open questions on subdire
t
sums of P -matri
es. Throughout the paper we give examples whi
h help illustrate
the theoreti
al results.
∗ Re
eived by the editors 16 February 2005. A
epted for publi
ation 22 June 2005. Handling
Editor: Mi
hael Neumann.
† Institut de Matemàti
a Multidis
iplinar, Universitat Politè
ni
a de Valèn
ia. Camí de Vera s/n.
46022 Valèn
ia. Spain (rbrumat.upv.es, pedro
hemat.upv.es). Supported by Spanish DGI and
FEDER grant MTM2004-02998 and by the O
ina de Cié
ia i Te
nologia de la Presidé
ia de La
Generalitat Valen
iana under proje
t GRUPOS03/062.
‡ Department of Mathemati
s, Temple University, Philadelphia, PA 19122-6094, U.S.A.
(szyldmath.temple.edu). Supported in part by the U.S. National S
ien
e Foundation under grant
DMS-0207525.
162
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
163
and of Their Inverses
2. Subdire
t sums of nonsingular matri
es. Let A and B be two square
matri
es of order n1 and n2 , respe
tively, and let k be an integer su
h that 1 ≤ k ≤
min(n1 , n2 ). Let A and B be partitioned into 2 × 2 blo
ks as follows:
A=
A11
A21
A12
A22
,
B=
B11
B21
B12
B22
(2.1)
,
where A22 and B11 are square matri
es of order k. Following [3℄, we
all the following
square matrix of order n = n1 + n2 − k,
A11
C = A21
0
A12
A22 + B11
B21
0
B12
B22
(2.2)
the k-subdire
t sum of A and B and denote it by C = A ⊕k B .
We are interested in the
ase when A and B are nonsingular matri
es. We partition the inverses of A and B
onformably to (2.1) and denote its blo
ks as follows:
A−1 =
Aˆ11
Aˆ21
Aˆ12
Aˆ22
,
B −1 =
ˆ11
B
ˆ21
B
ˆ12
B
ˆ22
B
(2.3)
,
where, as before, Aˆ22 and Bˆ11 are square of order k.
In the following result we show that nonsingularity of matrix Aˆ22 + Bˆ11 is a
ne
essary and su
ient
ondition for the k-subdire
t sum C to be nonsingular. The
proof is based on the use of the relation n = n1 + n2 − k to properly partition the
indi
ated matri
es.
Theorem 2.1.
spe
tively, and let
partitioned as in
Then
C
Let
A
and
B
be nonsingular matri
es of order
be an integer su
h that
(2.1) and
1 ≤ k ≤ min(n1 , n2 ).
their inverses be partitioned as in
(2.3).
n1
Let
Let
and n2 , reA and B be
C = A ⊕k B .
ˆ = Aˆ22 + B
ˆ11 is nonsingular.
H
be the identity matrix of order m. The theorem follows from the
is nonsingular if and only if
Proof. Let Im
following relation:
k
A−1
O
O
In−n1
C
In−n2
O
O
B −1
In−n2
= O
O
Aˆ12
ˆ
H
O
O
ˆ12 .
B
(2.4)
In−n1
M -matri
es. Given A = {aij } ∈ Rm×n , we write A > O
(A ≥ O), to indi
ate aij > 0 (aij ≥ 0), for i = 1, . . . , m, j = 1, . . . , n, and su
h
matri
es are
alled positive (nonnegative). Similarly, A ≥ B when A−B ≥ O. Square
matri
es whi
h have nonpositive o-diagonal entries are
alled Z -matri
es. We
all a
Z -matrix M a nonsingular M -matrix if M −1 ≥ O. We re
all some properties of these
2.1. Nonsingular
matri
es; see [1℄, [8℄:
(i) The diagonal of a nonsingular M -matrix is positive.
(ii) If B is a Z -matrix and M is a nonsingular M -matrix, and M ≤ B , then B is also
a nonsingular M -matrix. In parti
ular, any matrix obtained from a nonsingular M matrix by setting
ertain o-diagonal entries to zero is also a nonsingular M -matrix.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
164
R. Bru, F. Pedro
he, and D.B. Szyld
(iii) A matrix M is a nonsingular M -matrix if and only if ea
h prin
ipal submatrix
of M is a nonsingular M -matrix.
(iv) A Z -matrix M is a nonsingular M -matrix if and only if there exists a positive
ve
tor x > 0 su
h that M x > 0.
We rst
onsider the k-subdire
t sum of nonsingular Z -matri
es. >From (2.4) we
an expli
itly write
C −1 =
In−n2
O
O
B −1
ˆ −1 B
ˆ12 −1
Aˆ12 H
ˆ12 A
ˆ −1 B
−H
O
In−n1
ˆ −1
−Aˆ12 H
−1
ˆ
H
O
In−n2
O
O
O
In−n1
from whi
h we obtain
C −1
ˆ
ˆ −1 Aˆ21
A11 − Aˆ12 H
−1
ˆ
ˆ
=
B11 H Aˆ21
ˆ −1 Aˆ21
ˆ21 H
B
ˆ −1 Aˆ22
Aˆ12 − Aˆ12 H
−1 ˆ
ˆ
ˆ
B11 H A22
ˆ −1 Aˆ22
ˆ21 H
B
ˆ −1 B
ˆ12
Aˆ12 H
ˆ −1 B
ˆ12 + B
ˆ11 H
ˆ12
−B
−1 ˆ
ˆ
ˆ
ˆ
−B21 H B12 + B22
(2.5)
and therefore we
an state the following immediate result.
Z -matri
es of order n1 and n2 ,
1 ≤ k ≤ min(n1 , n2 ). Let A and B be
partitioned as in (2.1) and their inverses be partitioned as in (2.3). Let C = A ⊕k B .
ˆ = Aˆ22 + B
ˆ11 be nonsingular. Then C is a nonsingular M -matrix if and only if
Let H
−1
ea
h of the nine blo
ks of C
in (2.5) is nonnegative.
We
onsider now the
ase where A and B are nonsingular M -matri
es. It was
shown in [3℄ that even if H = A22 + B11 is a nonsingular M -matrix, this does not
guarantee that C = A⊕k B is a nonsingular M -matrix. We point out that this matrix
ˆ obtained from A−1 and B −1 and used in Theorem 2.1. The
H is not the matrix H
fa
t that H is a nonsingular M -matrix is a ne
essary but not a su
ient
ondition for
C to be a nonsingular M -matrix. Su
ient
onditions are presented in the following
Theorem 2.2.
Let
k
respe
tively, and let
A
and
B
be nonsingular
be an integer su
h that
result.
Let
Theorem 2.3. Let A and B be nonsingular M -matri
es partitioned as in (2.1).
x1 > 0 ∈ R(n1 −k)×1 , y1 > 0 ∈ Rk×1 , x2 > 0 ∈ Rk×1 and y2 > 0 ∈ R(n2 −k)×1 be
su
h that
A
Let
H = A22 + B11
x1
y1
> 0,
be a nonsingular
B
M -matrix
x2
y2
(2.6)
> 0.
and let
(2.7)
y = H −1 (A22 y1 + B11 x2 ) .
Then if
y ≤ y1
and
y ≤ x2
the
k -subdire
t
sum
C = A ⊕k B
is a nonsingular
M-
matrix.
Proof. We will show that there exists u > 0 su
h that Cu > 0. We rst note that
from (2.6) we get
A11 x1 + A12 y1
A21 x1 + A22 y1
> 0
> 0
,
B11 x2 + B12 y2
B21 x2 + B22 y2
>
>
0
0
.
(2.8)
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
and of Their Inverses
x1
Taking u = y and partitioning C as in (2.2) we obtain
y2
A11 x1 + A12 y
Cu = A21 x1 + (A22 + B11 )y + B12 y2 .
B21 y + B22 y2
165
(2.9)
Sin
e A21 ≤ O and B12 ≤ O, from (2.8) it follows that A22 y1 > 0 and B11 x2 > 0.
Sin
e H −1 ≥ O, from (2.7) we have that y is positive, and
onsequently, so is u, i.e.,
u > 0. We will show that Cu > 0 one blo
k of rows in (2.9) at a time. If y ≤ y1 ,
as A12 ≤ 0, we have that A12 y ≥ A12 y1 and again using (2.8) we obtain that the
rst blo
k of rows of Cu is positive. In a similar way, the
ondition y ≤ x2 together
with the last equation of (2.8) allows to
on
lude that the third blo
k of rows of
Cu is positive. Finally, substituting y given by (2.7) in the se
ond row of Cu and
onsidering (2.8) we
on
lude that the se
ond blo
k of rows of Cu is also positive.
Note that A and B are nonsingular M -matri
es and therefore the positive ve
tors (x1 , y1 ) and (x2 , y2 ) of (2.6) always exist. This theorem gives su
ient but not
ne
essary
onditions for C = A ⊕k B to be a nonsingular M -matrix, as illustrated in
Example 2.5 further below.
Example 2.4. The matri
es
−2 −1
3
A = −1/2 2 −3
−1 −1 4
and the ve
tors
x1
y1
1.8
= 2
1
1
−2 −1/3
9
0 ,
and B = −3
−2 −1/2
6
and
x2
y2
2.5
= 1
1
satisfy the inequalities (2.6), and
omputing the ve
tor y from (2.7) we get y ≈
(1.95, 0.87)T , whi
h satisfy y ≤ y1 and y ≤ x2 . Therefore the 2-subdire
t sum
−2 −1
0
3
−1/2 3
−5 −1/3
C =
−1 −4
13
0
0
−2 −1/2
6
is a nonsingular M -matrix in a
ordan
e with Theorem 2.3.
Example 2.5. The matri
es
5 −1/2 −1/3
A = −1
4
−2
10
−1 −6
and the ve
tors
x1
y1
1
= 1
1
1
−2 −1/3
0
9
and B = −3
−2 −1/2
6
and
x2
y2
2.5
= 1
1
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
166
R. Bru, F. Pedro
he, and D.B. Szyld
satisfy the inequalities (2.6), but
omputing ve
tor y from (2.7) we obtain
y ≈ (1.18, 0.85)T , whi
h does not satisfy the
onditions of Theorem 2.3. Nevertheless the 2-subdire
t sum
0
5 −1/2 −1/3
−1
5
−4 −1/3
C = A ⊕2 B =
−1 −9
19
0
0
−2 −1/2
6
is a nonsingular M -matrix.
In the spe
ial
ase of A and B blo
k lower and upper triangular nonsingular
M -matri
es, respe
tively, the results of Theorems 2.2 and 2.3 are easy to establish.
Let
A=
A11
A21
0
A22
,
B=
B11
0
B12
B22
(2.10)
,
with A22 and B11 square matri
es of order k.
Theorem 2.6.
Let
M -matri
es,
nonsingular M -matrix.
nonsingular
a
A
and
B
be nonsingular lower and upper blo
k triangular
respe
tively, partitioned as in
(2.10).
Then
C = A ⊕k B
is
Proof. We
an repeat the same argument as in the proof of Theorem 2.3 with
the advantage of having A12 = O and B21 = O. Note that
onditions y ≤ y1 and
y ≤ x2 are not ne
essary here be
ause the rst and last blo
k of rows of Cu in (2.9)
are automati
ally positive in this
ase.
−1
Remark 2.7. The expression of C
is given by (2.5). In this parti
ular
ase of
−1
ˆ
ˆ
blo
k triangular matri
es we have A12 = O, Bˆ21 = O, Aˆ22 = A−1
22 , B11 = B11 , from
ˆ = A−1 + B −1 . If, in addition, A22 = B11 , then we obtain
whi
h H
22
11
C −1
A−1
11
−1
= − 12 A22 A21 A−1
11
O
Example 2.8.
O
1 −1
A
2 22
O
O
−1
≥ O.
− 21 A−1
22 B12 B22
−1
B22
The matri
es
0
0
3
A = −1 5 −1
−1 −9 5
6
and B = −4
0
−2 −1
3 −3
0
2
satisfy the hypotheses of Theorem 2.6. The matri
es C = A ⊕2 B and C −1 are
3
0
0
0
−1 11 −3 −1
C =
−1 −13 8 −3 ,
0
0
0
2
C −1
0
0
0
1/3
11/147 8/49 3/49 17/98
=
8/49 13/49 11/49 23/49
0
0
0
1/2
and therefore C is a nonsingular M -matrix as expe
ted.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
167
and of Their Inverses
A and
A22 = B11 . In the next
we show that even if A and B are nonsingular M -matri
es, and so is the
blo
k, we
an not ensure that C = A ⊕k B is a nonsingular M -matrix.
In some appli
ations, su
h as in domain de
omposition [6℄, [7℄, matri
es
B
partitioned as in (2.1) arise with a
ommon blo
k, i.e.,
example
ommon
Example 2.9. The matri
es
370 −342 −318
A = −448 737 −107 ,
−46 −190 444
737 −107 −134
B = −190 444 −440
−885 −182 603
are nonsingular
M -matrix,
M -matri
es with A22 = B11
an
M -matrix,
but
C = A ⊕2 B
is not an
sin
e we have
370 −342 −318
0
−448 1474 −214 −134
C=
−46 −380 888 −440
0
−885 −182 603
C −1
and
−0.0291
−0.0145
≈
−0.0214
−0.0277
−0.0242
−0.0109
−0.0163
−0.0210
−0.0203
−0.0096
.
−0.0133
−0.0164
−0.0204
−0.0098
−0.0132
−0.0183
A and B share a blo
k and they are
M -matrix, the resulting k -subdire
t sum is in fa
t
In the next se
tion we shall see that when
submatri
es of a given nonsingular
a nonsingular
M -matrix.
2.2. Overlapping
M -matri
es.
A and B to be prinM -matrix and su
h that they have a
ommon
In this se
tion we restri
t
ipal submatri
es of a given nonsingular
blo
k. Let
M11
M = M21
M31
be a nonsingular
be of order
n1
M12
M22
M32
M13
M23
M33
(2.11)
M -matrix with M22 square matrix of order k ≥ 1
M11 M12
M22 M23
A=
and B =
M21 M22
M32 M33
and
n2 ,
respe
tively. The
k -subdire
t sum
M11
C = A ⊕k B = M21
O
In the following theorem we show that
C
M12
2M22
M32
of
and let
A and B
(2.12)
is thus given by
O
M23 .
M33
is a nonsingular
M -matrix.
(2.13)
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
168
R. Bru, F. Pedro
he, and D.B. Szyld
Let M be a nonsingular M -matrix partitioned as in (2.11),
B be two overlapping prin
ipal submatri
es given by (2.12). Then the
k -subdire
t sum C = A ⊕k B is a nonsingular M -matrix.
Proof. Let us
onstru
t an n × n Z -matrix T as follows:
M11 2M12 M13
T = M21 2M22 M23 .
(2.14)
M31 2M32 M33
Theorem 2.10.
and let
A
and
T = M diag(I, 2I, I) and we get T −1 = diag(I, (1/2)I, I)M −1 ≥ O. Then T is
a nonsingular M -matrix. Finally sin
e C is a Z -matrix and C ≥ T we
on
lude that
C is a nonsingular M -matrix.
Example 2.11. The following nonsingular M -matrix is partitioned as in (2.11):
13/14 −4/23 −3/20 −1/42 −19/186 −3/46
−3/7
21/23
−1/5 −1/21 −1/93
−6/23
−1/7 −7/46 17/20 −1/14 −1/186 −2/23
.
M =
(2.15)
−4/21 −27/92 −1/15
4/7
−58/93 −27/92
−1/14 −9/46 −3/10 −1/7
53/62
−9/46
−2/21 −9/92 −2/15 −2/7
−7/62
83/92
Then
Taking overlapping submatri
es
A and B
as in (2.12) the
3-subdire
t sum C = A⊕3 B
is given by
C=
13/14 −4/23 −3/20 −1/42 −19/186
0
−1/5 −1/21 −1/93
0
−3/7
21/23
−1/7 −7/46 17/10 −1/7
−1/93
−2/23
−4/21 −27/92 −2/15
8/7
−116/93 −27/92
53/31
−9/46
−1/14 −9/46 −3/5 −2/7
0
0
−2/15 −2/7
−7/62
83/92
and it is a nonsingular
C −1
2.3.
≈
M -matrix
1.3500
0.7628
0.3007
1.1024
0.4854
0.4543
k -subdire
t
a
ording to Theorem 2.10. In fa
t, we have that
0.3977
1.4108
0.2845
1.1571
0.5256
0.4743
sum of
0.2624
0.3383
0.7422
0.8927
0.5116
0.4564
0.1609
0.2085
0.2006
1.6092
0.4379
0.5941
p M -matri
es.
0.2103
0.2185
0.1824
1.3118
0.9664
0.5634
0.1232
0.1478
0.1763
0.8940
0.4013
1.4679
.
In this se
tion we extend Theo-
rems 2.3 and 2.10 to the subdire
t sum of several nonsingular
M -matri
es.
Exam-
ple 2.14 later in the se
tion illustrates the notation used in the proofs.
Theorem 2.12. Let
Ai ∈ Rni ×ni , i = 1, . . . p,
be nonsingular
M -matri
es
parti-
tioned as
Ai =
Ai,11
Ai,21
Ai,12
Ai,22
(2.16)
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
and of Their Inverses
169
with Ai,11 a square matrix of order ki−1 ≥ 1 and Ai,22 a square matrix of order ki ≥ 1,
i.e., ni = ki−1 + ki . Sin
e Ai are nonsingular M -matri
es we have that there exist
xi > 0 ∈ R(ni −ki )×1 and yi > 0 ∈ Rki ×1 su
h that
Ai
xi
yi
> 0,
i = 1, . . . , p.
(2.17)
Let C0 = A1 and de ne the following p − 1 ki -subdire
t sums:
Ci = Ci−1 ⊕ki Ai+1 ,
i = 1, . . . , p − 1,
(2.18)
i.e.,
C1
C2
=
=
A1 ⊕k1 A2 ,
(A1 ⊕k1 A2 ) ⊕k2 A3 = C1 ⊕k2 A3 ,
Cp−1
=
A1 ⊕k1 A2 ⊕k2 · · · ⊕kp−2 Ap−1 ⊕kp−1 Ap = Cp−2 ⊕kp −1 Ap .
..
.
Ea
h subdire
t sum Ci is of order mi , su
h that m0 = n1 and
mi = mi−1 + ni+1 − ki = mi−1 + ki+1 ,
i = 1, . . . , p − 1.
Let us partition Ci in the form
Ci =
Ci,11
Ci,21
Ci,12
Ci,22
,
i = 1, . . . , p − 1,
(2.19)
with Ci,22 a square matrix of order ki+1 . Let
Hi = Ci−1,22 + Ai+1,11 ,
i = 1, . . . p − 1,
be nonsingular M -matri
es and let
zi = Hi−1 (Ci−1,22 yi + Ai+1,11 xi+1 ),
i = 1, . . . p − 1.
Then, if zi ≤ yi and zi ≤ xi+1 , the subdire
t sums Ci given by (2.18) are nonsingular
M -matri
es for i = 1, . . . , p − 1.
Proof. It is easy to see that applying Theorem 2.3 to ea
h
onse
utive pair of
matri
es Ci we have that C1 , C2 , . . . , Cp−1 are nonsingular M -matri
es. This
an be
shown by indu
tion.
We now extend Theorem 2.10 to the sub-dire
t sum of p submatri
es of a given
nonsingular M -matrix M . To that end, we rst de ne M (S) a prin
ipal submatrix of
M with rows and
olumns with indi
es in the set of indi
es S = {i, i + 1, i + 2, . . . , j}.
In [2℄ we
all these
onse
utive prin
ipal submatri
es. For example, matri
es A and B
given by (2.12)
an be expressed as a submatri
es of M given by (2.11) as A = M (S1 ),
B = M (S2 ) with S1 = {1, 2} and S2 = {2, 3}.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
170
R. Bru, F. Pedro
he, and D.B. Szyld
Theorem 2.13.
1, . . . , p,
Let
M
M -matrix. Let Ai = M (Si ), i =
M and
onsider the p − 1 ki -subdire
t
be a nonsingular
be prin
ipal
onse
utive submatri
es of
sums given by
Ci = Ci−1 ⊕ki Ai+1 ,
in whi
h
C0 = A1 .
Then ea
h of the
(2.20)
i = 1, . . . , p − 1,
ki -subdire
t sums Ci
is a nonsingular
M -matrix.
. It is easy to relate the stru
ture of ea
h Ci to that of the submatri
es
Ai involved. We
onsider that Ai are overlapping prin
ipal submatri
es of the form
(2.12) but allowing that ea
h Ai has dierent number of blo
ks. Let M be partitioned
as
Proof
M =
M11
M21
M31
M12
M22
M32
M13
M23
M33
···
···
···
Mn1
Mn2
Mn3
···
..
.
..
.
..
.
..
M1n
M2n
M3n
..
.
.
Mnn
(2.21)
a
ording with the size of the prin
ipal submatri
es Ai . Ea
h blo
k Mij may be a
submatrix of more than one Am , m = 1, . . . , p. Let b(l)
ij ≥ 0 be the number of matri
es
Am su
h that Mij is a submatrix of Am , for m = 1, . . . , l + 1. Of
ourse we
an have
(l)
bij = 0. Let us
onsider the lth subdire
t sum Cl , 1 ≤ l ≤ p − 1, whi
h is of the form
Cl =
(l)
b11 M11
(l)
b21 M21
(l)
b31 M31
(l)
..
.
bl1 Ml1
(l)
b12 M12
(l)
b22 M22
(l)
b32 M32
(l)
..
.
bl2 Ml2
(l)
b13 M13
(l)
b23 M23
(l)
b33 M33
(l)
..
.
bl3 Ml3
···
···
···
..
(l)
b1l M1l
(l)
b2l M2l
(l)
b3l M3l
..
.
.
···
(l)
bll Mll
.
(2.22)
Observe that Cl is a Z -matrix and that b(l)
ii > 0. Furthermore, for ea
h
olumn it
(l)
(l)
holds that bii ≥ bji , j = 1, . . . , l.
The proof pro
eeds in a manner similar to that of Theorem 2.10. Consider the
Z -matrix (partitioned in the same manner as M )
(l)
(l)
(l)
(l)
Tl = Ml diag(b11 I, b22 I, b33 I, . . . , bll I),
where Ml is the prin
ipal submatrix of (2.21) with row and
olumn blo
ks from 1 to l.
It follows that Tl−1 ≥ O and therefore Tl is a nonsingular M -matrix. Finally, sin
e
Cl ≥ Tl , we
on
lude that Cl is a nonsingular M -matrix, l = 1, . . . , p.
Example 2.14. Given the nonsingular M -matrix M of Example 2.11, let us
onsider the following overlapping blo
ks
13/14 −4/23 −3/20
A1 = M ({1, 2, 3}) = −3/7 21/23 −1/5 ,
−1/7 −7/46 17/20
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
and of Their Inverses
171
21/23
−1/5 −1/21 −1/93
−7/46 17/20 −1/14 −1/186
,
A2 = M ({2, 3, 4, 5}) =
−27/92 −1/15
4/7
−58/93
−9/46 −3/10 −1/7
53/62
17/20 −1/14 −1/186 −2/23
−1/15
4/7
−58/93 −27/92
.
A3 = M ({3, 4, 5, 6}) =
−3/10 −1/7
53/62
−9/46
−2/15 −2/7 −7/62
83/92
2-subdire
t sum
13/14 −4/23 −3/20
0
0
−3/7 42/23
−2/5 −1/21 −1/93
C1 = A1 ⊕2 A2 = −1/7 −7/23 17/10 −1/14 −1/186
,
0
−27/92 −1/15
4/7
−58/93
0
−9/46 −3/10 −1/7
53/62
Then we have the
whi
h is a nonsingular
M -matrix,
C2 = C1 ⊕3 A3 =
and the
3-subdire
t
sum
13/14 −4/23 −3/20
0
0
0
−3/7 42/23
−2/5 −1/21 −1/93
0
−1/7 −7/23 51/20 −1/7
−1/93
−2/23
,
0
−27/92 −2/15
8/7
−116/93 −27/92
0
−9/46 −3/5 −2/7
53/31
−9/46
0
0
−2/15 −2/7
−7/62
83/92
M -matrix in a
ordan
e with Theorem 2.13. Observe that
k1 = 2 and k2 = 3. Note also that, for example, we have
(2)
(3)
(2)
(2)
= 0, b22 = 2, b22 = 2, b33 = 3, b14 = 0.
whi
h is also a nonsingular
in this example we have
(1)
(1)
(1)
b22 = 2, b33 = 2, b14
3. Subdire
t sums of inverses. Let
A
and
tioned as in (2.1). In this se
tion we
onsider the
B be nonsingular matri
es partik -subdire
t sum of their inverses.
We will establish
ounterparts to some of results in the previous se
tions.
denote by
G = A−1 ⊕k B −1 ,
with
A−1
ˆ
A11
G = Aˆ21
0
and
B −1
Let us
partitioned as in (2.3), i.e.,
Aˆ12
ˆ11
Aˆ22 + B
ˆ21
B
0
ˆ12 .
B
ˆ22
B
(3.1)
As a
orollary to, and in analogy to Theorem 2.1, the next statement indi
ates
A22 + B11 is a ne
essary
ondition to obtain G nonsingular.
and B be nonsingular matri
es partitioned as in (2.1) and
−1
let their inverses be partitioned as in (2.3). Let G = A
⊕k B −1 partitioned as in
(3.1) with k ≥ 1. Then G is nonsingular if and only if H = A22 +B11 is nonsingular.
that the nonsingularity of
Theorem 3.1. Let
A
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
172
R. Bru, F. Pedro
he, and D.B. Szyld
We remark that in analogy to the expression (2.5) of
G
C −1 ,
the expli
it form of
is
−1
G−1
A11 − A12 H −1 A21
=
B11 H −1 A21
B21 H −1 A21
Corollary 3.2. When
blo
k
A22 = B11
H = 2A22
and
B
A11
A21
A12
A22
k,
,
M -matri
es
(3.2)
with the
ommon
i.e., of the form
B=
A22
B21
B12
B22
,
(3.3)
G = A−1 ⊕k B −1 is nonsingular.
A and B are overlapping submatri
es
is nonsingular and therefore
We note that this is the
ase when
M -matrix, i.e.,
A12 H −1 B12
−B11 H −1 B12 + B12 .
−B21 H −1 B12 + B22
are nonsingular
a square matrix of order
A=
then
A
A12 − A12 H −1 A22
B11 H −1 A22
B21 H −1 A22
of an
of the form (2.12) and (2.11)
onsidered in Se
tion 2.2, where we were
interested in the subdire
t sum of
A and B .
Here we
on
lude that the subdire
t sum
of their inverses is always nonsingular.
A and B be the matri
es of
3-subdire
t sum of the inverses
1.5033 0.5513 0.5547
0.9540 1.5996 0.7158
0.6004 0.5636 2.9750
G = A−1 ⊕3 B −1 ≈
2.0383 2.1242 3.5729
0.8953 0.9650 2.0470
0
0
0.8551
Example 3.3. Let
Example 2.11, then a
ording to
Corollary 3.2, the
0.2757
0.3635
0.8144
6.5498
1.8025
1.3803
0.3912
0.4038
0.7407
5.3372
3.9062
1.2652
0
0
0.3708
2.0139
0.9048
1.9143
is a nonsingular matrix.
In the above example a dire
t
omputation shows that
G−1
≈
0.8900
−0.4682
−0.0714
−0.0952
−0.0357
0.1242
−0.2337
0.8566
−0.0761
−0.1467
−0.0978
0.2045
−0.0750
−0.1000
0.4250
−0.0333
−0.1500
−0.0667
−0.0119
−0.0238
−0.0357
0.2857
−0.0714
−0.1429
G−1
M -matrix:
0.0512
0.0470
−0.0435
−0.1467
−0.0978
0.7123
is not an
−0.0511
−0.0054
−0.0027
−0.3118
0.4274
−0.0565
Z -matrix. Note that when A and B are M -matri
es we have from
G = A−1 ⊕ B −1 is nonnegative. Therefore assuming that G−1 exists we
have (G−1 )−1 ≥ O. Then it is a natural question to seek
onditions so that G−1 is a
nonsingular M -matrix. We study this question next.
The expressions (3.1) of G and (3.2) of G−1 , Theorem 3.1, and the observation
that for nonsingular M -matri
es we have (G−1 )−1 ≥ O, imply the following result.
Theorem 3.4. Let A and B be nonsingular M -matri
es partitioned as in (2.1)
−1
⊕k B −1 with k ≥ 1, and let
and their inverses partitioned as in (2.3). Let G = A
H = A22 + B11 be nonsingular. Then G−1 is a nonsingular M -matrix if and only if
G−1 is a Z -matrix.
whi
h is not a
(3.1) that
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
173
and of Their Inverses
Corollary 3.5. Let A and B be lower and upper blo
k triangular nonsingular
M -matri
es, respe
tively, partitioned as in (2.10) with A22 and B11 square matri
es
−1
of order k and H = A22 + B11 nonsingular. Then G
= (A−1 ⊕k B −1 )−1 is a
nonsingular M -matrix if and only if the following
onditions hold:
i) B11 H −1 A21 ≤ O.
ii) B11 H −1 A22 is a Z -matrix.
iii) −B11 H −1 B12 + B12 ≤ O.
Proof.
>From (3.2) and (2.10) we have that
−1
G
and therefore
0
B11 H −1 A22
0
A11
−1
B
H
A21
=
11
0
G−1
Z -matrix
is a
0
−B11 H −1 B12 + B12
B22
(3.4)
if and only if the
onditions i), ii) and iii) hold.
Conditions i), ii) and iii) in the
orollary are not as stringent as they may appear.
For example, let
A
B
and
A=
Then
M -matri
es partitioned as
A22 = B11 , a square matrix of order k , i.e.,
0
A22 B12
.
and B =
(3.5)
A22
0
B22
be blo
k triangular nonsingular
in (2.10) with a
ommon blo
k
A11
A21
G−1 = (A−1 ⊕k B −1 )−1
is a nonsingular
M -matrix,
sin
e we have from (3.4)
that
G−1
and therefore
G−1
is a
A11
= 12 A21
O
Z -matrix.
O
1
A
2 22
O
O
1
,
2 B12
B22
In fa
t, in this
ase, we have
A−1
11
−1
G = −A22 A21 A−1
11
O
O
2A−1
22
O
O
−1
≥ O.
−A−1
22 B12 B22
−1
B22
The next example illustrates this situation.
Example 3.6. Let
A
and
B
G = A−1 ⊕2 B −1
be the matri
es of Example 2.8, then
0
0
0
1/3
1/8 49/80 21/80 9/20
=
7/24 77/80 73/80 11/10 ,
0
0
0
1/2
and
G−1
0
0
0
3
−18/49 146/49 −6/7 −39/49
=
−4/7
−22/7
2
−11/7
0
0
0
2
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
174
R. Bru, F. Pedro
he, and D.B. Szyld
is a nonsingular M -matrix in a
ordan
e with Corollary 3.5.
Note that if the hypotheses of Corollary 3.5 are satis ed, and re
alling Theorem 2.6, we have that ea
h of the matri
es C = A ⊕k B and G−1 = (A−1 ⊕k B −1 )−1
are both nonsingular M -matri
es.
4. P -matri
es. A square matrix is a P -matrix if all its prin
ipal minors are
positive. As a
onsequen
e we have that all the diagonal entries of a P -matrix are
positive. It is also follows that a nonsingular M -matrix is a P -matrix. It
an also be
shown that if A is a nonsingular M -matrix, then A−1 is a P -matrix; see, e.g., [5℄.
In [3℄ it is shown that the k-subdire
t sum (with k > 1) of two P -matri
es is
not ne
essarily a P -matrix. Our results in Se
tions 2.1 and 3 hold for nonsingular
M -matri
es and inverses of M -matri
es, respe
tively. As these two
lasses of matri
es
are subsets of P -matri
es, it is natural to ask if similar su
ient
onditions
an be
found so that the k-subdire
t sum of P -matri
es is a P -matrix. The following example
indi
ates that the answer may not be easy to obtain, sin
e even in the simplest
ase
of diagonal submatri
es the k-subdire
t sum may not be a P -matrix.
Example 4.1. Given the P -matri
es
543 388 322
A = 69 160 0 ,
368 0 375
we have that the 2-subdire
t sum
136 0 219
B = 0 225 159
61 177 230
543 388 322 0
69 296 0 219
C = A ⊕2 B =
368 0 600 159
0
61 177 230
is not a P -matrix, sin
e det(C) < 0.
A
knowledgment. We thank the referee for a very
areful reading of the manus
ript
and for his
omments.
REFERENCES
[1℄ A. Berman and R.J. Plemmons. Nonnegative Matri
es in the Mathemati
al S
ien
es, A
ademi
Press, New York, 1979. Reprinted and updated, SIAM, Philadelphia, 1994.
[2℄ R. Bru, F. Pedro
he, and D.B. Szyld. Additive S
hwarz Iterations for Markov Chains, SIAM
J. Matrix Anal. Appl., to appear.
[3℄ S.M. Fallat and C.R. Johnson. Sub-dire
t sums and positivity
lasses of matri
es. Linear Algebra
Appl., 288:149173, 1999.
[4℄ A. Frommer and D.B. Szyld. Weighted Max Norms, Splittings,and Overlapping Additive
S
hwarz Iterations, Numeris
he Mathematik, 83:259278, 1999.
[5℄ R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, New York, 1985.
[6℄ B.F. Smith, P.E. Bjørstad, and W.D. Gropp. Domain de
omposition: Parallel multilevel methods for ellipti
partial dierential equations. Cambridge University Press, Cambridge, 1996.
[7℄ A. Toselli and O.B. Widlund. Domain De
omposition: Algorithms and Theory. Springer Series
in Computational Mathemati
s, vol. 34. Springer, New York, 2005.
[8℄ R.S. Varga. Matrix Iterative Analysis. Prenti
e-Hall, Englewood Clis, New Jersey, 1962.
Se
ond Edition, revised and expanded, Springer, New York, 2000.
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
SUBDIRECT SUMS OF NONSINGULAR
M -MATRICES
AND OF
THEIR INVERSES∗
RAFAEL BRU† , FRANCISCO PEDROCHE† ,
AND
DANIEL B. SZYLD‡
Abstra
t. The question of when the subdire
t sum of two nonsingular M -matri
es is a nonsingular M -matrix is studied. Su
ient
onditions are given. The
ase of inverses of M -matri
es is
also studied. In parti
ular, it is shown that the subdire
t sum of overlapping prin
ipal submatri
es
of a nonsingular M -matrix is a nonsingular M -matrix. Some examples illustrating the
onditions
presented are also given.
AMS subje
t
lassi
ations. 15A48.
Key words. Subdire
t sum, M -matri
es, Inverse of M -matrix, Overlapping blo
ks.
1. Introdu
tion. Subdire
t sum of matri
es are generalizations of the usual sum
of matri
es (a k-subdire
t sum is formally de ned below in Se
tion 2). They were
introdu
ed by Fallat and Johnson in [3℄, where many of their properties were analyzed.
For example, they showed that the subdire
t sum of positive de nite matri
es, or of
symmetri
M -matri
es, are positive de nite or symmetri
M -matri
es, respe
tively.
They also showed that this is not the
ase for M -matri
es: the sum of two M -matri
es
may not be an M -matrix. One goal of the present paper is to give su
ient
onditions
so that the subdire
t sum of nonsingular M -matri
es is a nonsingular M -matrix. We
also treat the
ase of the subdire
t sum of inverses of M -matri
es.
Subdire
t sums of two overlapping prin
ipal submatri
es of a nonsingular M matrix appear naturally when analyzing additive S
hwarz methods for Markov
hains
or other matri
es [2℄, [4℄. In this paper we show that the subdire
t sum of two
overlapping prin
ipal submatri
es of a nonsingular M -matrix is a nonsingular M matrix.
The paper is stru
tured as follows. In Se
tion 2 we fo
us on the nonsingularity of
the subdire
t sum of any pair of nonsingular matri
es, giving an expli
it expression
for the inverse. In Se
tion 2.1 we study the k-subdire
t sum of two nonsingular
M -matri
es and in parti
ular, the
ase of subdire
t sums of overlapping blo
ks of
nonsingular M -matri
es. In Se
tion 2.3 we extend some results to the subdire
t sum
of more than two nonsingular M -matri
es. In Se
tion 3 we analyze the subdire
t sum
of two inverses. Finally, in Se
tion 4 we mention some open questions on subdire
t
sums of P -matri
es. Throughout the paper we give examples whi
h help illustrate
the theoreti
al results.
∗ Re
eived by the editors 16 February 2005. A
epted for publi
ation 22 June 2005. Handling
Editor: Mi
hael Neumann.
† Institut de Matemàti
a Multidis
iplinar, Universitat Politè
ni
a de Valèn
ia. Camí de Vera s/n.
46022 Valèn
ia. Spain (rbrumat.upv.es, pedro
hemat.upv.es). Supported by Spanish DGI and
FEDER grant MTM2004-02998 and by the O
ina de Cié
ia i Te
nologia de la Presidé
ia de La
Generalitat Valen
iana under proje
t GRUPOS03/062.
‡ Department of Mathemati
s, Temple University, Philadelphia, PA 19122-6094, U.S.A.
(szyldmath.temple.edu). Supported in part by the U.S. National S
ien
e Foundation under grant
DMS-0207525.
162
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
163
and of Their Inverses
2. Subdire
t sums of nonsingular matri
es. Let A and B be two square
matri
es of order n1 and n2 , respe
tively, and let k be an integer su
h that 1 ≤ k ≤
min(n1 , n2 ). Let A and B be partitioned into 2 × 2 blo
ks as follows:
A=
A11
A21
A12
A22
,
B=
B11
B21
B12
B22
(2.1)
,
where A22 and B11 are square matri
es of order k. Following [3℄, we
all the following
square matrix of order n = n1 + n2 − k,
A11
C = A21
0
A12
A22 + B11
B21
0
B12
B22
(2.2)
the k-subdire
t sum of A and B and denote it by C = A ⊕k B .
We are interested in the
ase when A and B are nonsingular matri
es. We partition the inverses of A and B
onformably to (2.1) and denote its blo
ks as follows:
A−1 =
Aˆ11
Aˆ21
Aˆ12
Aˆ22
,
B −1 =
ˆ11
B
ˆ21
B
ˆ12
B
ˆ22
B
(2.3)
,
where, as before, Aˆ22 and Bˆ11 are square of order k.
In the following result we show that nonsingularity of matrix Aˆ22 + Bˆ11 is a
ne
essary and su
ient
ondition for the k-subdire
t sum C to be nonsingular. The
proof is based on the use of the relation n = n1 + n2 − k to properly partition the
indi
ated matri
es.
Theorem 2.1.
spe
tively, and let
partitioned as in
Then
C
Let
A
and
B
be nonsingular matri
es of order
be an integer su
h that
(2.1) and
1 ≤ k ≤ min(n1 , n2 ).
their inverses be partitioned as in
(2.3).
n1
Let
Let
and n2 , reA and B be
C = A ⊕k B .
ˆ = Aˆ22 + B
ˆ11 is nonsingular.
H
be the identity matrix of order m. The theorem follows from the
is nonsingular if and only if
Proof. Let Im
following relation:
k
A−1
O
O
In−n1
C
In−n2
O
O
B −1
In−n2
= O
O
Aˆ12
ˆ
H
O
O
ˆ12 .
B
(2.4)
In−n1
M -matri
es. Given A = {aij } ∈ Rm×n , we write A > O
(A ≥ O), to indi
ate aij > 0 (aij ≥ 0), for i = 1, . . . , m, j = 1, . . . , n, and su
h
matri
es are
alled positive (nonnegative). Similarly, A ≥ B when A−B ≥ O. Square
matri
es whi
h have nonpositive o-diagonal entries are
alled Z -matri
es. We
all a
Z -matrix M a nonsingular M -matrix if M −1 ≥ O. We re
all some properties of these
2.1. Nonsingular
matri
es; see [1℄, [8℄:
(i) The diagonal of a nonsingular M -matrix is positive.
(ii) If B is a Z -matrix and M is a nonsingular M -matrix, and M ≤ B , then B is also
a nonsingular M -matrix. In parti
ular, any matrix obtained from a nonsingular M matrix by setting
ertain o-diagonal entries to zero is also a nonsingular M -matrix.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
164
R. Bru, F. Pedro
he, and D.B. Szyld
(iii) A matrix M is a nonsingular M -matrix if and only if ea
h prin
ipal submatrix
of M is a nonsingular M -matrix.
(iv) A Z -matrix M is a nonsingular M -matrix if and only if there exists a positive
ve
tor x > 0 su
h that M x > 0.
We rst
onsider the k-subdire
t sum of nonsingular Z -matri
es. >From (2.4) we
an expli
itly write
C −1 =
In−n2
O
O
B −1
ˆ −1 B
ˆ12 −1
Aˆ12 H
ˆ12 A
ˆ −1 B
−H
O
In−n1
ˆ −1
−Aˆ12 H
−1
ˆ
H
O
In−n2
O
O
O
In−n1
from whi
h we obtain
C −1
ˆ
ˆ −1 Aˆ21
A11 − Aˆ12 H
−1
ˆ
ˆ
=
B11 H Aˆ21
ˆ −1 Aˆ21
ˆ21 H
B
ˆ −1 Aˆ22
Aˆ12 − Aˆ12 H
−1 ˆ
ˆ
ˆ
B11 H A22
ˆ −1 Aˆ22
ˆ21 H
B
ˆ −1 B
ˆ12
Aˆ12 H
ˆ −1 B
ˆ12 + B
ˆ11 H
ˆ12
−B
−1 ˆ
ˆ
ˆ
ˆ
−B21 H B12 + B22
(2.5)
and therefore we
an state the following immediate result.
Z -matri
es of order n1 and n2 ,
1 ≤ k ≤ min(n1 , n2 ). Let A and B be
partitioned as in (2.1) and their inverses be partitioned as in (2.3). Let C = A ⊕k B .
ˆ = Aˆ22 + B
ˆ11 be nonsingular. Then C is a nonsingular M -matrix if and only if
Let H
−1
ea
h of the nine blo
ks of C
in (2.5) is nonnegative.
We
onsider now the
ase where A and B are nonsingular M -matri
es. It was
shown in [3℄ that even if H = A22 + B11 is a nonsingular M -matrix, this does not
guarantee that C = A⊕k B is a nonsingular M -matrix. We point out that this matrix
ˆ obtained from A−1 and B −1 and used in Theorem 2.1. The
H is not the matrix H
fa
t that H is a nonsingular M -matrix is a ne
essary but not a su
ient
ondition for
C to be a nonsingular M -matrix. Su
ient
onditions are presented in the following
Theorem 2.2.
Let
k
respe
tively, and let
A
and
B
be nonsingular
be an integer su
h that
result.
Let
Theorem 2.3. Let A and B be nonsingular M -matri
es partitioned as in (2.1).
x1 > 0 ∈ R(n1 −k)×1 , y1 > 0 ∈ Rk×1 , x2 > 0 ∈ Rk×1 and y2 > 0 ∈ R(n2 −k)×1 be
su
h that
A
Let
H = A22 + B11
x1
y1
> 0,
be a nonsingular
B
M -matrix
x2
y2
(2.6)
> 0.
and let
(2.7)
y = H −1 (A22 y1 + B11 x2 ) .
Then if
y ≤ y1
and
y ≤ x2
the
k -subdire
t
sum
C = A ⊕k B
is a nonsingular
M-
matrix.
Proof. We will show that there exists u > 0 su
h that Cu > 0. We rst note that
from (2.6) we get
A11 x1 + A12 y1
A21 x1 + A22 y1
> 0
> 0
,
B11 x2 + B12 y2
B21 x2 + B22 y2
>
>
0
0
.
(2.8)
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
and of Their Inverses
x1
Taking u = y and partitioning C as in (2.2) we obtain
y2
A11 x1 + A12 y
Cu = A21 x1 + (A22 + B11 )y + B12 y2 .
B21 y + B22 y2
165
(2.9)
Sin
e A21 ≤ O and B12 ≤ O, from (2.8) it follows that A22 y1 > 0 and B11 x2 > 0.
Sin
e H −1 ≥ O, from (2.7) we have that y is positive, and
onsequently, so is u, i.e.,
u > 0. We will show that Cu > 0 one blo
k of rows in (2.9) at a time. If y ≤ y1 ,
as A12 ≤ 0, we have that A12 y ≥ A12 y1 and again using (2.8) we obtain that the
rst blo
k of rows of Cu is positive. In a similar way, the
ondition y ≤ x2 together
with the last equation of (2.8) allows to
on
lude that the third blo
k of rows of
Cu is positive. Finally, substituting y given by (2.7) in the se
ond row of Cu and
onsidering (2.8) we
on
lude that the se
ond blo
k of rows of Cu is also positive.
Note that A and B are nonsingular M -matri
es and therefore the positive ve
tors (x1 , y1 ) and (x2 , y2 ) of (2.6) always exist. This theorem gives su
ient but not
ne
essary
onditions for C = A ⊕k B to be a nonsingular M -matrix, as illustrated in
Example 2.5 further below.
Example 2.4. The matri
es
−2 −1
3
A = −1/2 2 −3
−1 −1 4
and the ve
tors
x1
y1
1.8
= 2
1
1
−2 −1/3
9
0 ,
and B = −3
−2 −1/2
6
and
x2
y2
2.5
= 1
1
satisfy the inequalities (2.6), and
omputing the ve
tor y from (2.7) we get y ≈
(1.95, 0.87)T , whi
h satisfy y ≤ y1 and y ≤ x2 . Therefore the 2-subdire
t sum
−2 −1
0
3
−1/2 3
−5 −1/3
C =
−1 −4
13
0
0
−2 −1/2
6
is a nonsingular M -matrix in a
ordan
e with Theorem 2.3.
Example 2.5. The matri
es
5 −1/2 −1/3
A = −1
4
−2
10
−1 −6
and the ve
tors
x1
y1
1
= 1
1
1
−2 −1/3
0
9
and B = −3
−2 −1/2
6
and
x2
y2
2.5
= 1
1
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
166
R. Bru, F. Pedro
he, and D.B. Szyld
satisfy the inequalities (2.6), but
omputing ve
tor y from (2.7) we obtain
y ≈ (1.18, 0.85)T , whi
h does not satisfy the
onditions of Theorem 2.3. Nevertheless the 2-subdire
t sum
0
5 −1/2 −1/3
−1
5
−4 −1/3
C = A ⊕2 B =
−1 −9
19
0
0
−2 −1/2
6
is a nonsingular M -matrix.
In the spe
ial
ase of A and B blo
k lower and upper triangular nonsingular
M -matri
es, respe
tively, the results of Theorems 2.2 and 2.3 are easy to establish.
Let
A=
A11
A21
0
A22
,
B=
B11
0
B12
B22
(2.10)
,
with A22 and B11 square matri
es of order k.
Theorem 2.6.
Let
M -matri
es,
nonsingular M -matrix.
nonsingular
a
A
and
B
be nonsingular lower and upper blo
k triangular
respe
tively, partitioned as in
(2.10).
Then
C = A ⊕k B
is
Proof. We
an repeat the same argument as in the proof of Theorem 2.3 with
the advantage of having A12 = O and B21 = O. Note that
onditions y ≤ y1 and
y ≤ x2 are not ne
essary here be
ause the rst and last blo
k of rows of Cu in (2.9)
are automati
ally positive in this
ase.
−1
Remark 2.7. The expression of C
is given by (2.5). In this parti
ular
ase of
−1
ˆ
ˆ
blo
k triangular matri
es we have A12 = O, Bˆ21 = O, Aˆ22 = A−1
22 , B11 = B11 , from
ˆ = A−1 + B −1 . If, in addition, A22 = B11 , then we obtain
whi
h H
22
11
C −1
A−1
11
−1
= − 12 A22 A21 A−1
11
O
Example 2.8.
O
1 −1
A
2 22
O
O
−1
≥ O.
− 21 A−1
22 B12 B22
−1
B22
The matri
es
0
0
3
A = −1 5 −1
−1 −9 5
6
and B = −4
0
−2 −1
3 −3
0
2
satisfy the hypotheses of Theorem 2.6. The matri
es C = A ⊕2 B and C −1 are
3
0
0
0
−1 11 −3 −1
C =
−1 −13 8 −3 ,
0
0
0
2
C −1
0
0
0
1/3
11/147 8/49 3/49 17/98
=
8/49 13/49 11/49 23/49
0
0
0
1/2
and therefore C is a nonsingular M -matrix as expe
ted.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
167
and of Their Inverses
A and
A22 = B11 . In the next
we show that even if A and B are nonsingular M -matri
es, and so is the
blo
k, we
an not ensure that C = A ⊕k B is a nonsingular M -matrix.
In some appli
ations, su
h as in domain de
omposition [6℄, [7℄, matri
es
B
partitioned as in (2.1) arise with a
ommon blo
k, i.e.,
example
ommon
Example 2.9. The matri
es
370 −342 −318
A = −448 737 −107 ,
−46 −190 444
737 −107 −134
B = −190 444 −440
−885 −182 603
are nonsingular
M -matrix,
M -matri
es with A22 = B11
an
M -matrix,
but
C = A ⊕2 B
is not an
sin
e we have
370 −342 −318
0
−448 1474 −214 −134
C=
−46 −380 888 −440
0
−885 −182 603
C −1
and
−0.0291
−0.0145
≈
−0.0214
−0.0277
−0.0242
−0.0109
−0.0163
−0.0210
−0.0203
−0.0096
.
−0.0133
−0.0164
−0.0204
−0.0098
−0.0132
−0.0183
A and B share a blo
k and they are
M -matrix, the resulting k -subdire
t sum is in fa
t
In the next se
tion we shall see that when
submatri
es of a given nonsingular
a nonsingular
M -matrix.
2.2. Overlapping
M -matri
es.
A and B to be prinM -matrix and su
h that they have a
ommon
In this se
tion we restri
t
ipal submatri
es of a given nonsingular
blo
k. Let
M11
M = M21
M31
be a nonsingular
be of order
n1
M12
M22
M32
M13
M23
M33
(2.11)
M -matrix with M22 square matrix of order k ≥ 1
M11 M12
M22 M23
A=
and B =
M21 M22
M32 M33
and
n2 ,
respe
tively. The
k -subdire
t sum
M11
C = A ⊕k B = M21
O
In the following theorem we show that
C
M12
2M22
M32
of
and let
A and B
(2.12)
is thus given by
O
M23 .
M33
is a nonsingular
M -matrix.
(2.13)
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
168
R. Bru, F. Pedro
he, and D.B. Szyld
Let M be a nonsingular M -matrix partitioned as in (2.11),
B be two overlapping prin
ipal submatri
es given by (2.12). Then the
k -subdire
t sum C = A ⊕k B is a nonsingular M -matrix.
Proof. Let us
onstru
t an n × n Z -matrix T as follows:
M11 2M12 M13
T = M21 2M22 M23 .
(2.14)
M31 2M32 M33
Theorem 2.10.
and let
A
and
T = M diag(I, 2I, I) and we get T −1 = diag(I, (1/2)I, I)M −1 ≥ O. Then T is
a nonsingular M -matrix. Finally sin
e C is a Z -matrix and C ≥ T we
on
lude that
C is a nonsingular M -matrix.
Example 2.11. The following nonsingular M -matrix is partitioned as in (2.11):
13/14 −4/23 −3/20 −1/42 −19/186 −3/46
−3/7
21/23
−1/5 −1/21 −1/93
−6/23
−1/7 −7/46 17/20 −1/14 −1/186 −2/23
.
M =
(2.15)
−4/21 −27/92 −1/15
4/7
−58/93 −27/92
−1/14 −9/46 −3/10 −1/7
53/62
−9/46
−2/21 −9/92 −2/15 −2/7
−7/62
83/92
Then
Taking overlapping submatri
es
A and B
as in (2.12) the
3-subdire
t sum C = A⊕3 B
is given by
C=
13/14 −4/23 −3/20 −1/42 −19/186
0
−1/5 −1/21 −1/93
0
−3/7
21/23
−1/7 −7/46 17/10 −1/7
−1/93
−2/23
−4/21 −27/92 −2/15
8/7
−116/93 −27/92
53/31
−9/46
−1/14 −9/46 −3/5 −2/7
0
0
−2/15 −2/7
−7/62
83/92
and it is a nonsingular
C −1
2.3.
≈
M -matrix
1.3500
0.7628
0.3007
1.1024
0.4854
0.4543
k -subdire
t
a
ording to Theorem 2.10. In fa
t, we have that
0.3977
1.4108
0.2845
1.1571
0.5256
0.4743
sum of
0.2624
0.3383
0.7422
0.8927
0.5116
0.4564
0.1609
0.2085
0.2006
1.6092
0.4379
0.5941
p M -matri
es.
0.2103
0.2185
0.1824
1.3118
0.9664
0.5634
0.1232
0.1478
0.1763
0.8940
0.4013
1.4679
.
In this se
tion we extend Theo-
rems 2.3 and 2.10 to the subdire
t sum of several nonsingular
M -matri
es.
Exam-
ple 2.14 later in the se
tion illustrates the notation used in the proofs.
Theorem 2.12. Let
Ai ∈ Rni ×ni , i = 1, . . . p,
be nonsingular
M -matri
es
parti-
tioned as
Ai =
Ai,11
Ai,21
Ai,12
Ai,22
(2.16)
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
and of Their Inverses
169
with Ai,11 a square matrix of order ki−1 ≥ 1 and Ai,22 a square matrix of order ki ≥ 1,
i.e., ni = ki−1 + ki . Sin
e Ai are nonsingular M -matri
es we have that there exist
xi > 0 ∈ R(ni −ki )×1 and yi > 0 ∈ Rki ×1 su
h that
Ai
xi
yi
> 0,
i = 1, . . . , p.
(2.17)
Let C0 = A1 and de ne the following p − 1 ki -subdire
t sums:
Ci = Ci−1 ⊕ki Ai+1 ,
i = 1, . . . , p − 1,
(2.18)
i.e.,
C1
C2
=
=
A1 ⊕k1 A2 ,
(A1 ⊕k1 A2 ) ⊕k2 A3 = C1 ⊕k2 A3 ,
Cp−1
=
A1 ⊕k1 A2 ⊕k2 · · · ⊕kp−2 Ap−1 ⊕kp−1 Ap = Cp−2 ⊕kp −1 Ap .
..
.
Ea
h subdire
t sum Ci is of order mi , su
h that m0 = n1 and
mi = mi−1 + ni+1 − ki = mi−1 + ki+1 ,
i = 1, . . . , p − 1.
Let us partition Ci in the form
Ci =
Ci,11
Ci,21
Ci,12
Ci,22
,
i = 1, . . . , p − 1,
(2.19)
with Ci,22 a square matrix of order ki+1 . Let
Hi = Ci−1,22 + Ai+1,11 ,
i = 1, . . . p − 1,
be nonsingular M -matri
es and let
zi = Hi−1 (Ci−1,22 yi + Ai+1,11 xi+1 ),
i = 1, . . . p − 1.
Then, if zi ≤ yi and zi ≤ xi+1 , the subdire
t sums Ci given by (2.18) are nonsingular
M -matri
es for i = 1, . . . , p − 1.
Proof. It is easy to see that applying Theorem 2.3 to ea
h
onse
utive pair of
matri
es Ci we have that C1 , C2 , . . . , Cp−1 are nonsingular M -matri
es. This
an be
shown by indu
tion.
We now extend Theorem 2.10 to the sub-dire
t sum of p submatri
es of a given
nonsingular M -matrix M . To that end, we rst de ne M (S) a prin
ipal submatrix of
M with rows and
olumns with indi
es in the set of indi
es S = {i, i + 1, i + 2, . . . , j}.
In [2℄ we
all these
onse
utive prin
ipal submatri
es. For example, matri
es A and B
given by (2.12)
an be expressed as a submatri
es of M given by (2.11) as A = M (S1 ),
B = M (S2 ) with S1 = {1, 2} and S2 = {2, 3}.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
170
R. Bru, F. Pedro
he, and D.B. Szyld
Theorem 2.13.
1, . . . , p,
Let
M
M -matrix. Let Ai = M (Si ), i =
M and
onsider the p − 1 ki -subdire
t
be a nonsingular
be prin
ipal
onse
utive submatri
es of
sums given by
Ci = Ci−1 ⊕ki Ai+1 ,
in whi
h
C0 = A1 .
Then ea
h of the
(2.20)
i = 1, . . . , p − 1,
ki -subdire
t sums Ci
is a nonsingular
M -matrix.
. It is easy to relate the stru
ture of ea
h Ci to that of the submatri
es
Ai involved. We
onsider that Ai are overlapping prin
ipal submatri
es of the form
(2.12) but allowing that ea
h Ai has dierent number of blo
ks. Let M be partitioned
as
Proof
M =
M11
M21
M31
M12
M22
M32
M13
M23
M33
···
···
···
Mn1
Mn2
Mn3
···
..
.
..
.
..
.
..
M1n
M2n
M3n
..
.
.
Mnn
(2.21)
a
ording with the size of the prin
ipal submatri
es Ai . Ea
h blo
k Mij may be a
submatrix of more than one Am , m = 1, . . . , p. Let b(l)
ij ≥ 0 be the number of matri
es
Am su
h that Mij is a submatrix of Am , for m = 1, . . . , l + 1. Of
ourse we
an have
(l)
bij = 0. Let us
onsider the lth subdire
t sum Cl , 1 ≤ l ≤ p − 1, whi
h is of the form
Cl =
(l)
b11 M11
(l)
b21 M21
(l)
b31 M31
(l)
..
.
bl1 Ml1
(l)
b12 M12
(l)
b22 M22
(l)
b32 M32
(l)
..
.
bl2 Ml2
(l)
b13 M13
(l)
b23 M23
(l)
b33 M33
(l)
..
.
bl3 Ml3
···
···
···
..
(l)
b1l M1l
(l)
b2l M2l
(l)
b3l M3l
..
.
.
···
(l)
bll Mll
.
(2.22)
Observe that Cl is a Z -matrix and that b(l)
ii > 0. Furthermore, for ea
h
olumn it
(l)
(l)
holds that bii ≥ bji , j = 1, . . . , l.
The proof pro
eeds in a manner similar to that of Theorem 2.10. Consider the
Z -matrix (partitioned in the same manner as M )
(l)
(l)
(l)
(l)
Tl = Ml diag(b11 I, b22 I, b33 I, . . . , bll I),
where Ml is the prin
ipal submatrix of (2.21) with row and
olumn blo
ks from 1 to l.
It follows that Tl−1 ≥ O and therefore Tl is a nonsingular M -matrix. Finally, sin
e
Cl ≥ Tl , we
on
lude that Cl is a nonsingular M -matrix, l = 1, . . . , p.
Example 2.14. Given the nonsingular M -matrix M of Example 2.11, let us
onsider the following overlapping blo
ks
13/14 −4/23 −3/20
A1 = M ({1, 2, 3}) = −3/7 21/23 −1/5 ,
−1/7 −7/46 17/20
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
and of Their Inverses
171
21/23
−1/5 −1/21 −1/93
−7/46 17/20 −1/14 −1/186
,
A2 = M ({2, 3, 4, 5}) =
−27/92 −1/15
4/7
−58/93
−9/46 −3/10 −1/7
53/62
17/20 −1/14 −1/186 −2/23
−1/15
4/7
−58/93 −27/92
.
A3 = M ({3, 4, 5, 6}) =
−3/10 −1/7
53/62
−9/46
−2/15 −2/7 −7/62
83/92
2-subdire
t sum
13/14 −4/23 −3/20
0
0
−3/7 42/23
−2/5 −1/21 −1/93
C1 = A1 ⊕2 A2 = −1/7 −7/23 17/10 −1/14 −1/186
,
0
−27/92 −1/15
4/7
−58/93
0
−9/46 −3/10 −1/7
53/62
Then we have the
whi
h is a nonsingular
M -matrix,
C2 = C1 ⊕3 A3 =
and the
3-subdire
t
sum
13/14 −4/23 −3/20
0
0
0
−3/7 42/23
−2/5 −1/21 −1/93
0
−1/7 −7/23 51/20 −1/7
−1/93
−2/23
,
0
−27/92 −2/15
8/7
−116/93 −27/92
0
−9/46 −3/5 −2/7
53/31
−9/46
0
0
−2/15 −2/7
−7/62
83/92
M -matrix in a
ordan
e with Theorem 2.13. Observe that
k1 = 2 and k2 = 3. Note also that, for example, we have
(2)
(3)
(2)
(2)
= 0, b22 = 2, b22 = 2, b33 = 3, b14 = 0.
whi
h is also a nonsingular
in this example we have
(1)
(1)
(1)
b22 = 2, b33 = 2, b14
3. Subdire
t sums of inverses. Let
A
and
tioned as in (2.1). In this se
tion we
onsider the
B be nonsingular matri
es partik -subdire
t sum of their inverses.
We will establish
ounterparts to some of results in the previous se
tions.
denote by
G = A−1 ⊕k B −1 ,
with
A−1
ˆ
A11
G = Aˆ21
0
and
B −1
Let us
partitioned as in (2.3), i.e.,
Aˆ12
ˆ11
Aˆ22 + B
ˆ21
B
0
ˆ12 .
B
ˆ22
B
(3.1)
As a
orollary to, and in analogy to Theorem 2.1, the next statement indi
ates
A22 + B11 is a ne
essary
ondition to obtain G nonsingular.
and B be nonsingular matri
es partitioned as in (2.1) and
−1
let their inverses be partitioned as in (2.3). Let G = A
⊕k B −1 partitioned as in
(3.1) with k ≥ 1. Then G is nonsingular if and only if H = A22 +B11 is nonsingular.
that the nonsingularity of
Theorem 3.1. Let
A
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
172
R. Bru, F. Pedro
he, and D.B. Szyld
We remark that in analogy to the expression (2.5) of
G
C −1 ,
the expli
it form of
is
−1
G−1
A11 − A12 H −1 A21
=
B11 H −1 A21
B21 H −1 A21
Corollary 3.2. When
blo
k
A22 = B11
H = 2A22
and
B
A11
A21
A12
A22
k,
,
M -matri
es
(3.2)
with the
ommon
i.e., of the form
B=
A22
B21
B12
B22
,
(3.3)
G = A−1 ⊕k B −1 is nonsingular.
A and B are overlapping submatri
es
is nonsingular and therefore
We note that this is the
ase when
M -matrix, i.e.,
A12 H −1 B12
−B11 H −1 B12 + B12 .
−B21 H −1 B12 + B22
are nonsingular
a square matrix of order
A=
then
A
A12 − A12 H −1 A22
B11 H −1 A22
B21 H −1 A22
of an
of the form (2.12) and (2.11)
onsidered in Se
tion 2.2, where we were
interested in the subdire
t sum of
A and B .
Here we
on
lude that the subdire
t sum
of their inverses is always nonsingular.
A and B be the matri
es of
3-subdire
t sum of the inverses
1.5033 0.5513 0.5547
0.9540 1.5996 0.7158
0.6004 0.5636 2.9750
G = A−1 ⊕3 B −1 ≈
2.0383 2.1242 3.5729
0.8953 0.9650 2.0470
0
0
0.8551
Example 3.3. Let
Example 2.11, then a
ording to
Corollary 3.2, the
0.2757
0.3635
0.8144
6.5498
1.8025
1.3803
0.3912
0.4038
0.7407
5.3372
3.9062
1.2652
0
0
0.3708
2.0139
0.9048
1.9143
is a nonsingular matrix.
In the above example a dire
t
omputation shows that
G−1
≈
0.8900
−0.4682
−0.0714
−0.0952
−0.0357
0.1242
−0.2337
0.8566
−0.0761
−0.1467
−0.0978
0.2045
−0.0750
−0.1000
0.4250
−0.0333
−0.1500
−0.0667
−0.0119
−0.0238
−0.0357
0.2857
−0.0714
−0.1429
G−1
M -matrix:
0.0512
0.0470
−0.0435
−0.1467
−0.0978
0.7123
is not an
−0.0511
−0.0054
−0.0027
−0.3118
0.4274
−0.0565
Z -matrix. Note that when A and B are M -matri
es we have from
G = A−1 ⊕ B −1 is nonnegative. Therefore assuming that G−1 exists we
have (G−1 )−1 ≥ O. Then it is a natural question to seek
onditions so that G−1 is a
nonsingular M -matrix. We study this question next.
The expressions (3.1) of G and (3.2) of G−1 , Theorem 3.1, and the observation
that for nonsingular M -matri
es we have (G−1 )−1 ≥ O, imply the following result.
Theorem 3.4. Let A and B be nonsingular M -matri
es partitioned as in (2.1)
−1
⊕k B −1 with k ≥ 1, and let
and their inverses partitioned as in (2.3). Let G = A
H = A22 + B11 be nonsingular. Then G−1 is a nonsingular M -matrix if and only if
G−1 is a Z -matrix.
whi
h is not a
(3.1) that
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
Subdire
t Sums of Nonsingular
M -matri
es
173
and of Their Inverses
Corollary 3.5. Let A and B be lower and upper blo
k triangular nonsingular
M -matri
es, respe
tively, partitioned as in (2.10) with A22 and B11 square matri
es
−1
of order k and H = A22 + B11 nonsingular. Then G
= (A−1 ⊕k B −1 )−1 is a
nonsingular M -matrix if and only if the following
onditions hold:
i) B11 H −1 A21 ≤ O.
ii) B11 H −1 A22 is a Z -matrix.
iii) −B11 H −1 B12 + B12 ≤ O.
Proof.
>From (3.2) and (2.10) we have that
−1
G
and therefore
0
B11 H −1 A22
0
A11
−1
B
H
A21
=
11
0
G−1
Z -matrix
is a
0
−B11 H −1 B12 + B12
B22
(3.4)
if and only if the
onditions i), ii) and iii) hold.
Conditions i), ii) and iii) in the
orollary are not as stringent as they may appear.
For example, let
A
B
and
A=
Then
M -matri
es partitioned as
A22 = B11 , a square matrix of order k , i.e.,
0
A22 B12
.
and B =
(3.5)
A22
0
B22
be blo
k triangular nonsingular
in (2.10) with a
ommon blo
k
A11
A21
G−1 = (A−1 ⊕k B −1 )−1
is a nonsingular
M -matrix,
sin
e we have from (3.4)
that
G−1
and therefore
G−1
is a
A11
= 12 A21
O
Z -matrix.
O
1
A
2 22
O
O
1
,
2 B12
B22
In fa
t, in this
ase, we have
A−1
11
−1
G = −A22 A21 A−1
11
O
O
2A−1
22
O
O
−1
≥ O.
−A−1
22 B12 B22
−1
B22
The next example illustrates this situation.
Example 3.6. Let
A
and
B
G = A−1 ⊕2 B −1
be the matri
es of Example 2.8, then
0
0
0
1/3
1/8 49/80 21/80 9/20
=
7/24 77/80 73/80 11/10 ,
0
0
0
1/2
and
G−1
0
0
0
3
−18/49 146/49 −6/7 −39/49
=
−4/7
−22/7
2
−11/7
0
0
0
2
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 13, pp. 162-174, July 2005
ELA
www.math.technion.ac.il/iic/ela
174
R. Bru, F. Pedro
he, and D.B. Szyld
is a nonsingular M -matrix in a
ordan
e with Corollary 3.5.
Note that if the hypotheses of Corollary 3.5 are satis ed, and re
alling Theorem 2.6, we have that ea
h of the matri
es C = A ⊕k B and G−1 = (A−1 ⊕k B −1 )−1
are both nonsingular M -matri
es.
4. P -matri
es. A square matrix is a P -matrix if all its prin
ipal minors are
positive. As a
onsequen
e we have that all the diagonal entries of a P -matrix are
positive. It is also follows that a nonsingular M -matrix is a P -matrix. It
an also be
shown that if A is a nonsingular M -matrix, then A−1 is a P -matrix; see, e.g., [5℄.
In [3℄ it is shown that the k-subdire
t sum (with k > 1) of two P -matri
es is
not ne
essarily a P -matrix. Our results in Se
tions 2.1 and 3 hold for nonsingular
M -matri
es and inverses of M -matri
es, respe
tively. As these two
lasses of matri
es
are subsets of P -matri
es, it is natural to ask if similar su
ient
onditions
an be
found so that the k-subdire
t sum of P -matri
es is a P -matrix. The following example
indi
ates that the answer may not be easy to obtain, sin
e even in the simplest
ase
of diagonal submatri
es the k-subdire
t sum may not be a P -matrix.
Example 4.1. Given the P -matri
es
543 388 322
A = 69 160 0 ,
368 0 375
we have that the 2-subdire
t sum
136 0 219
B = 0 225 159
61 177 230
543 388 322 0
69 296 0 219
C = A ⊕2 B =
368 0 600 159
0
61 177 230
is not a P -matrix, sin
e det(C) < 0.
A
knowledgment. We thank the referee for a very
areful reading of the manus
ript
and for his
omments.
REFERENCES
[1℄ A. Berman and R.J. Plemmons. Nonnegative Matri
es in the Mathemati
al S
ien
es, A
ademi
Press, New York, 1979. Reprinted and updated, SIAM, Philadelphia, 1994.
[2℄ R. Bru, F. Pedro
he, and D.B. Szyld. Additive S
hwarz Iterations for Markov Chains, SIAM
J. Matrix Anal. Appl., to appear.
[3℄ S.M. Fallat and C.R. Johnson. Sub-dire
t sums and positivity
lasses of matri
es. Linear Algebra
Appl., 288:149173, 1999.
[4℄ A. Frommer and D.B. Szyld. Weighted Max Norms, Splittings,and Overlapping Additive
S
hwarz Iterations, Numeris
he Mathematik, 83:259278, 1999.
[5℄ R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, New York, 1985.
[6℄ B.F. Smith, P.E. Bjørstad, and W.D. Gropp. Domain de
omposition: Parallel multilevel methods for ellipti
partial dierential equations. Cambridge University Press, Cambridge, 1996.
[7℄ A. Toselli and O.B. Widlund. Domain De
omposition: Algorithms and Theory. Springer Series
in Computational Mathemati
s, vol. 34. Springer, New York, 2005.
[8℄ R.S. Varga. Matrix Iterative Analysis. Prenti
e-Hall, Englewood Clis, New Jersey, 1962.
Se
ond Edition, revised and expanded, Springer, New York, 2000.