vol13_pp162-174. 208KB Jun 04 2011 12:05:53 AM

(1)

SUBDIRECT SUMSOFNONSINGULAR

M

-MATRICESAND OF THEIR INVERSES

∗

RAFAEL BRU

†

, FRANCISCO PEDROCHE

†

, AND DANIEL B.SZYLD

‡

Abstrat. Thequestionofwhenthe subdiretsumoftwo nonsingular

M

-matriesisa non-singular

M

-matrixisstudied. Suientonditionsaregiven. Theaseofinversesof

M

-matriesis alsostudied. Inpartiular,itisshownthatthesubdiretsumofoverlappingprinipalsubmatries ofa nonsingular

M

-matrixisanonsingular

M

-matrix. Someexamplesillustratingthe onditions presentedarealsogiven.

AMSsubjet lassiations.15A48.

Keywords.Subdiretsum,

M

-matries,Inverseof

M

-matrix,Overlappingbloks.

1. Introdution. Subdiretsumofmatriesaregeneralizationsoftheusualsum of matries (a

k

-subdiret sum is formally dened below in Setion 2). Theywere introduedbyFallatandJohnsonin[3℄,wheremanyoftheirpropertieswereanalyzed. Forexample,theyshowedthatthesubdiret sumofpositivedenitematries,orof symmetri

M

-matries, arepositivedenite orsymmetri

M

-matries,respetively. Theyalsoshowedthatthisisnottheasefor

M

-matries: thesumoftwo

M

-matries maynotbean

M

-matrix. Onegoalofthepresentpaperistogivesuientonditions sothatthesubdiretsumofnonsingular

M

-matriesisanonsingular

M

-matrix.We alsotreattheaseofthesubdiretsumofinversesof

M

-matries.

Subdiret sums of two overlapping prinipal submatries of a nonsingular

M

-matrixappearnaturallywhenanalyzingadditiveShwarzmethodsforMarkovhains or other matries [2℄, [4℄. In this paper we show that the subdiret sum of two overlapping prinipal submatries of a nonsingular

M

-matrix is a nonsingular

M

-matrix.

Thepaperisstruturedasfollows. InSetion2wefousonthenonsingularityof thesubdiret sum ofanypairof nonsingularmatries, givinganexpliit expression for the inverse. In Setion 2.1 we study the

k

-subdiret sum of two nonsingular

M

-matries and in partiular, the ase of subdiret sums of overlapping bloks of

nonsingular

M

-matries. InSetion2.3weextendsomeresultstothesubdiretsum ofmorethantwononsingular

M

-matries. InSetion3weanalyzethesubdiretsum of twoinverses. Finally, in Setion 4we mentionsomeopen questions on subdiret sums of

P

-matries. Throughout the paper we give exampleswhih help illustrate thetheoretialresults.

∗

Reeivedbytheeditors 16February 2005. Aepted forpubliation22June 2005. Handling Editor: MihaelNeumann.

†

InstitutdeMatemàtiaMultidisiplinar,UniversitatPolitèniadeValènia.CamídeVeras/n. 46022Valènia. Spain(rbrumat.upv.es, pedrohemat.upv.es). SupportedbySpanishDGIand FEDERgrantMTM2004-02998 andbythe Oina deCiéiaiTenologiadelaPresidéia deLa GeneralitatValenianaunderprojetGRUPOS03/062.

‡

(2)

2. Subdiret sums of nonsingular matries. Let

A

and

B

be two square matriesoforder

n

1

and

n

2

,respetively,andlet

k

beanintegersuhthat

1

≤

k

≤

min(

n

1

, n

2)

. Let

A

and

B

bepartitionedinto

2

×

2

bloksasfollows:

A

=

A

11

A

12

A

21

A

22

,

B

=

B

11

B

12

B

21

B

22

,

(2.1)

where

A

22

and

B

11

aresquarematriesoforder

k

. Following[3℄,weallthefollowing squarematrixof order

n

=

n

1

+

n

2

−

k

,

C

=





A

11

A

12

0

A

21

A

22

+

B

11

B

12

0

B

21

B

22





(2.2)

the

k

-subdiretsumof

A

and

B

anddenoteitby

C

=

A

⊕

k

B

.

Weareinterestedin theasewhen

A

and

B

arenonsingularmatries. We parti-tiontheinversesof

A

and

B

onformablyto(2.1)anddenoteitsbloksasfollows:

A

−

1

=

ˆ

A

11

A

ˆ

12

ˆ

A

21

A

ˆ

22

,

B

−

1

=

ˆ

B

11

B

ˆ

12

ˆ

B

21

B

ˆ

22

,

(2.3)

where,asbefore,

A

ˆ

22

and

B

ˆ

11

aresquareoforder

k

.

In the following result we show that nonsingularity of matrix

A

ˆ

22

+ ˆ

B

11

is a neessaryandsuientonditionforthe

k

-subdiret sum

C

tobenonsingular. The proof is basedon theuse of therelation

n

=

n

1

+

n

2

−

k

to properly partition the indiatedmatries.

Theorem 2.1. Let

A

and

B

be nonsingular matries of order

n

1

and

n

2

, re-spetively, and let

k

be an integer suhthat

1

≤

k

≤

min(

n

1

, n

2)

. Let

A

and

B

be partitionedasin(2.1)andtheirinverses bepartitionedasin(2.3). Let

C

=

A

⊕

k

B

. Then

C

isnonsingularif andonlyif

H

ˆ

= ˆ

A

22

+ ˆ

B

11

isnonsingular.

Proof. Let

I

m

betheidentitymatrixof order

m

. Thetheorem followsfrom the followingrelation:

A

−

1

O

I

n

−

n

1

C

I

n

−

n

2

O

B

−

1

=





I

n

−

n

2

A

ˆ

12

O

H

ˆ

B

ˆ

12

O

I

n

−

n

1





.

(2.4)

2.1. Nonsingular

M

-matries. Given

A

=

{

a

ij

} ∈

R

m

×

n

, wewrite

A > O

(

A

≥

O

), to indiate

a

ij

>

0

(

a

ij

≥

0

), for

i

= 1

, . . . , m

,

j

= 1

, . . . , n

, and suh matriesarealledpositive(nonnegative). Similarly,

A

≥

B

when

A

−

B

≥

O

. Square matrieswhihhavenonpositiveo-diagonalentriesarealled

Z

-matries. Wealla

Z

-matrix

M

anonsingular

M

-matrixif

M

−

1

≥

O

. Wereallsomepropertiesofthese matries;see[1℄,[8℄:

(i)Thediagonalofanonsingular

M

-matrixispositive.

(3)

(iii) Amatrix

M

is anonsingular

M

-matrixifand onlyif eah prinipal submatrix of

M

isanonsingular

M

-matrix.

(iv) A

Z

-matrix

M

is anonsingular

M

-matrixif and only ifthere exists apositive vetor

x >

0

suh that

M x >

0

.

Werstonsiderthe

k

-subdiretsumofnonsingular

Z

-matries. >From(2.4)we anexpliitlywrite

C

−

1

=

I

n

−

n

2

O

B

−

1





I

n

−

n

2

−

A

ˆ

12

H

ˆ

−

1

A

ˆ

12

H

ˆ

−

1

B

ˆ

12

O

H

ˆ

−

1

−

H

ˆ

−

1

B

ˆ

12

O

I

n

−

n

1





A

−

1

O

I

n

−

n

1

fromwhihweobtain

C

−

1

=





ˆ

A

11

−

A

ˆ

12

H

ˆ

−

1

A

ˆ

21

A

ˆ

12

−

A

ˆ

12

H

ˆ

−

1

A

ˆ

22

A

ˆ

12

H

ˆ

−

1

B

ˆ

12

ˆ

B

11

H

ˆ

−

1

A

ˆ

21

B

ˆ

11

H

ˆ

−

1

A

ˆ

22

−

B

ˆ

11

H

ˆ

−

1

B

ˆ

12

+ ˆ

B

12

ˆ

B

21

H

ˆ

−

1

A

ˆ

21

B

ˆ

21

H

ˆ

−

1

A

ˆ

22

−

B

ˆ

21

H

ˆ

−

1

B

ˆ

12

+ ˆ

B

22





(2.5)

andthereforeweanstatethefollowingimmediateresult.

Theorem 2.2. Let

A

and

B

be nonsingular

Z

-matries of order

n

1

and

n

2

, respetively,and let

k

be an integersuh that

1

≤

k

≤

min(

n

1

, n

2)

. Let

A

and

B

be partitionedasin(2.1)andtheir inversesbepartitionedasin(2.3). Let

C

=

A

⊕

k

B

. Let

H

ˆ

= ˆ

A

22

+ ˆ

B

11

benonsingular. Then

C

isanonsingular

M

-matrixifandonly if eah ofthe nine bloksof

C

−

1

in(2.5)isnonnegative.

We onsider nowthe ase where

A

and

B

are nonsingular

M

-matries. It was shown in [3℄ that even if

H

=

A

22

+

B

11

is a nonsingular

M

-matrix, this doesnot guaranteethat

C

=

A

⊕

k

B

isanonsingular

M

-matrix. Wepointoutthatthismatrix

H

is notthematrix

H

ˆ

obtainedfrom

A

−

1

and

B

−

1

andused in Theorem 2.1. The fatthat

H

isanonsingular

M

-matrixisaneessarybutnotasuientonditionfor

C

tobeanonsingular

M

-matrix. Suientonditionsarepresentedinthefollowing

result.

Theorem 2.3. Let

A

and

B

benonsingular

M

-matriespartitionedasin (2.1). Let

x

1

>

0

∈

R

(

n

1

−

k

)

×

1

,

y

1

>

0

∈

R

k

×

1

,

x

2

>

0

∈

R

k

×

1

and

y

2

>

0

∈

R

(

n

2

−

k

)

×

1

be suhthat

A

x

1

y

1

>

0

,

B

x

2

y

2

>

0

.

(2.6)

Let

H

=

A

22

+

B

11

beanonsingular

M

-matrixandlet

y

=

H

−

1

(

A

22

y

1

+

B

11

x

2)

.

(2.7)

Then if

y

≤

y

1

and

y

≤

x

2

the

k

-subdiret sum

C

=

A

⊕

k

B

is a nonsingular

M

-matrix.

Proof. Wewillshowthatthereexists

u >

0

suhthat

Cu >

0

. Werstnotethat from(2.6)weget

A

11

x

1

+

A

12

y

1

>

0

A

21

x

1

+

A

22

y

1

>

0

,

B

11

x

2

+

B

12

y

2

>

0

B

21

x

2

+

B

22

y

2

>

0

(4)

Taking

u

=





x

1

y

2





andpartitioning

C

asin(2.2)weobtain

Cu

=





A

11

x

1

+

A

12

y

A

21

x

1

+ (

A

22

+

B

11)

y

+

B

12

y

2

B

21

y

+

B

22

y

2





.

(2.9)

Sine

A

21

≤

O

and

B

12

≤

O

, from (2.8) it followsthat

A

22

y

1

>

0

and

B

11

x

2

>

0

. Sine

H

−

1

≥

O

, from(2.7)wehavethat

y

ispositive,and onsequently,sois

u

,i.e.,

u >

0

. We will show that

Cu >

0

oneblokof rowsin (2.9) at atime. If

y

≤

y

1

,

as

A

12

≤

0

, we have that

A

12

y

≥

A

12

y

1

and again using (2.8) weobtain that the rstblokofrowsof

Cu

ispositive. Inasimilarway, theondition

y

≤

x

2

together with the last equation of (2.8) allows to onlude that the third blok of rows of

Cu

is positive. Finally, substituting

y

given by(2.7) in the seond row of

Cu

and

onsidering(2.8)weonludethat theseondblokofrowsof

Cu

isalsopositive. Note that

A

and

B

arenonsingular

M

-matries and thereforethe positive ve-tors

(

x

1

, y

1)

and

(

x

2

, y

2)

of(2.6)alwaysexist. Thistheorem givessuientbut not neessaryonditionsfor

C

=

A

⊕

k

B

to beanonsingular

M

-matrix,asillustratedin Example2.5furtherbelow.

Example2.4. Thematries

A

=





3

−

2

−

1

−

1

/

2

−

3

−

1

−

1

4





and

B

=





1

−

2

−

1

/

3

−

3

9

0

−

2

−

1

/

2

6





,

andthevetors

x

1

y

1

=





1

.

8

2

1





and

x

2

y

2

=





2

.

5

1





satisfy the inequalities (2.6), and omputing the vetor

y

from (2.7) we get

y

≈

(1

.

95

,

0

.

87)

T

,whih satisfy

y

≤

y

1

and

y

≤

x

2

. Thereforethe

2

-subdiretsum

C

=







3

−

2

−

1

0

−

1

/

2

3

−

5

−

1

/

3

−

1

−

4

13

0

−

2

−

1

/

2

6







isanonsingular

M

-matrixinaordanewith Theorem2.3. Example2.5. Thematries

A

=





5

−

1

/

2

−

1

/

3

−

1

4

−

2

−

1

−

6

10





and

B

=





1

−

2

−

1

/

3

−

3

9

0

−

2

−

1

/

2

6





andthevetors

x

1

y

1

=





1





and

x

2

y

2

=





2

.

5

1



(5)

satisfytheinequalities(2.6),butomputingvetor

y

from(2.7)weobtain

y

≈

(1

.

18

,

0

.

85)

T

, whihdoesnotsatisfytheonditionsof Theorem2.3. Never-thelessthe

2

-subdiret sum

C

=

A

⊕

2

B

=







5

−

1

/

2

−

1

/

3

0

−

1

5

−

4

−

1

/

3

−

1

−

9

19

0

−

2

−

1

/

2

6







isanonsingular

M

-matrix.

In the speial ase of

A

and

B

blok lower and upper triangular nonsingular

M

-matries, respetively, theresultsof Theorems 2.2 and 2.3 areeasy to establish.

Let

A

=

A

11

0

A

21

A

22

,

B

=

B

11

B

12

0

B

22

,

(2.10)

with

A

22

and

B

11

squarematriesoforder

k

.

Theorem 2.6. Let

A

and

B

be nonsingular lower and upper blok triangular nonsingular

M

-matries, respetively,partitioned asin (2.10). Then

C

=

A

⊕

k

B

is anonsingular

M

-matrix.

Proof. We anrepeat the same argumentas in the proof of Theorem 2.3 with the advantage of having

A

12

=

O

and

B

21

=

O

. Note that onditions

y

≤

y

1

and

y

≤

x

2

arenotneessaryherebeausetherstand lastblokofrowsof

Cu

in (2.9) areautomatiallypositivein thisase.

Remark 2.7. Theexpressionof

C

−

1

isgivenby(2.5). Inthis partiularaseof bloktriangularmatrieswehave

A

ˆ

12

=

O

,

B

ˆ

21

=

O

,

A

ˆ

22

=

A

−

1

22

,

B

ˆ

11

=

B

−

1

11

,from

whih

H

ˆ

=

A

−

1

22

+

B

−

1

11

. If,inaddition,

A

22

=

B

11

,thenweobtain

C

−

1

=





A

−

1

11

O

−

1

2

A

−

1

22

A

21

A

−

11

1

2

A

−

1

22

−

1

2

A

−

1

22

B

12

B

22

−

1

O

B

−

1

22





≥

O.

Example 2.8. Thematries

A

=





3

0

−

1

5

−

1

−

1

−

9

5





and

B

=





6

−

2

−

1

−

4

3

−

3

0

2





satisfythehypothesesofTheorem 2.6. Thematries

C

=

A

⊕

2

B

and

C

−

1

are

C

=







3

0

−

1

11

−

3

−

1

−

1

−

13

8

−

3

0

2







,

C

−

1

=







1

/

3

0

11

/

147

8

/

49

3

/

49

17

/

98

8

/

49

13

/

49 11

/

49 23

/

49

0

1

/

2







(6)

Insome appliations, suh asin domain deomposition [6℄, [7℄, matries

A

and

B

partitioned as in (2.1)arise with aommon blok, i.e.,

A

22

=

B

11

. In the next

example we show that even if

A

and

B

are nonsingular

M

-matries, and so is the ommonblok,weannotensurethat

C

=

A

⊕

k

B

isanonsingular

M

-matrix.

Example2.9. Thematries

A

=





370

−

342

−

318

−

448

737

−

107

−

46

−

190

444





,

B

=





737

−

107

−

134

−

190

444

−

440

−

885

−

182

603





arenonsingular

M

-matrieswith

A

22

=

B

11

an

M

-matrix,but

C

=

A

⊕

2

B

isnotan

M

-matrix,sinewehave

C

=







370

−

342

−

318

0

−

448

1474

−

214

−

134

−

46

−

380

888

−

440

0

−

885

−

182

603







and

C

.

0164







.

Inthenext setionwe shallsee that when

A

and

B

share ablok andthey are submatriesofagivennonsingular

M

-matrix,theresulting

k

-subdiretsumisinfat anonsingular

M

-matrix.

2.2. Overlapping

M

-matries. Inthissetionwerestrit

A

and

B

tobe prin-ipalsubmatriesofagivennonsingular

M

-matrixandsuhthattheyhaveaommon blok. Let

M

=





M

11

M

12

M

13

M

21

M

22

M

23

M

31

M

32

M

33





(2.11)

beanonsingular

M

-matrixwith

M

22

squarematrixoforder

k

≥

1

andlet

A

=

M

11

M

12

M

21

M

22

and

B

=

M

22

M

23

M

32

M

33

(2.12)

beoforder

n

1

and

n

2

,respetively. The

k

-subdiretsumof

A

and

B

isthusgivenby

C

=

A

⊕

k

B

=





M

11

M

12

O

M

21

2

M

22

M

23

O

M

32

M

33





.

(2.13)

(7)

Theorem 2.10. Let

M

be a nonsingular

M

-matrix partitioned as in (2.11), andlet

A

and

B

be twooverlapping prinipal submatries given by (2.12). Then the

k

-subdiretsum

C

=

A

⊕

k

B

is anonsingular

M

-matrix. Proof. Letusonstrutan

n

×

n Z

-matrix

T

asfollows:

T

=





M

11

2

M

12

M

13

M

21

2

M

22

M

23

M

31

2

M

32

M

33





.

(2.14)

Then

T

=

M

diag(

I,

2

I, I

)

andweget

T

−

1

= diag(

I,

(1

/

2)

I, I

)

M

−

1

≥

O

. Then

T

is anonsingular

M

-matrix. Finallysine

C

isa

Z

-matrixand

C

≥

T

weonludethat

C

isanonsingular

M

-matrix.

Example2.11. Thefollowingnonsingular

M

-matrixispartitionedasin (2.11):

M

=







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

−

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







.

(2.15)

Takingoverlappingsubmatries

A

and

B

asin(2.12)the

3

-subdiretsum

C

=

A

⊕

3

B

isgivenby

C

=







13

/

14

−

4

/

23

−

3

/

20

−

1

/

42

−

19

/

186

0

−

3

/

7

21

/

23

−

1

/

5

−

1

/

21

−

1

/

93

0

−

1

/

7

−

7

/

46

17

/

10

−

1

/

7

−

1

/

93

−

2

/

23

−

4

/

21

−

27

/

92

−

2

/

15

8

/

7

−

116

/

93

−

27

/

92

−

1

/

14

−

9

/

46

−

3

/

5

−

2

/

7

53

/

31

−

9

/

46

0

−

2

/

15

−

2

/

7

−

7

/

62

83

/

92







anditisanonsingular

M

-matrixaordingtoTheorem2.10. Infat,wehavethat

C

−

1

≈







1

.

3500 0

.

3977 0

.

2624 0

.

1609 0

.

2103 0

.

1232

0

.

7628 1

.

4108 0

.

3383 0

.

2085 0

.

2185 0

.

1478

0

.

3007 0

.

2845 0

.

7422 0

.

2006 0

.

1824 0

.

1763

1

.

1024 1

.

1571 0

.

8927 1

.

6092 1

.

3118 0

.

8940

0

.

4854 0

.

5256 0

.

5116 0

.

4379 0

.

9664 0

.

4013

0

.

4543 0

.

4743 0

.

4564 0

.

5941 0

.

5634 1

.

4679







.

2.3.

k

-subdiret sum of

p M

-matries. In this setion we extend Theo-rems 2.3 and 2.10 to the subdiret sum of several nonsingular

M

-matries. Exam-ple2.14laterinthesetionillustratesthenotationusedin theproofs.

Theorem 2.12. Let

A

i

∈

R

n

i

×

n

i

,

i

= 1

, . . . p

,be nonsingular

M

-matries parti-tionedas

A

i

=

A

i,

11

A

i,

12

A

i,

21

A

i,

22

(8)

with

A

i,

11

asquarematrixoforder

k

i

−

1

≥

1

and

A

i,

22

asquarematrixoforder

k

i

≥

1

, i.e.,

n

i

=

k

i

−

1

+

k

i

. Sine

A

i

are nonsingular

M

-matries we have that there exist

x

i

>

0

∈

R

(

n

i

−

k

i

)

×

1

and

y

i

>

0

∈

R

k

i

×

1

suhthat

A

i

x

i

y

i

>

0

,

i

= 1

, . . . , p.

(2.17)

Let

C

0

=

A

1

anddene the following

p

−

1

k

i

-subdiretsums:

C

i

=

C

i

−

1

⊕

k

i

A

i

+1

,

i

= 1

, . . . , p

−

1

,

(2.18)

i.e.,

C

1

=

A

1

⊕

k

1

A

2

,

C

2

=

(

A

1

⊕

k

1

A

2)

⊕

k

2

A

3

=

C

1

⊕

k

2

A

3

,

. . .

C

p

−

1

=

A

1

⊕

k

1

A

2

⊕

k

2

· · · ⊕

k

p

−2

A

p

−

1

⊕

k

p

−1

A

p

=

C

p

−

2

⊕

k

p

−

1

A

p

.

Eahsubdiretsum

C

i

isof order

m

i

,suhthat

m

0

=

n

1

and

m

i

=

m

i

−

1

+

n

i

+1

−

k

i

=

m

i

−

1

+

k

i

+1

,

i

= 1

, . . . , p

−

1

.

Letuspartition

C

i

in the form

C

i

=

C

i,

11

C

i,

12

C

i,

21

C

i,

22

,

i

= 1

, . . . , p

−

1

,

(2.19)

with

C

i,

22

asquarematrixof order

k

i

+1

. Let

H

i

=

C

i

−

1

,

22

+

A

i

+1

,

11

,

i

= 1

, . . . p

−

1

,

benonsingular

M

-matriesandlet

z

i

=

H

i

−

1

(

C

i

−

1

,

22

y

i

+

A

i

+1

,

11

x

i

+1)

,

i

= 1

, . . . p

−

1

.

Then,if

z

i

≤

y

i

and

z

i

≤

x

i

+1

,thesubdiretsums

C

i

given by(2.18)are nonsingular

M

-matriesfor

i

= 1

, . . . , p

−

1

.

Proof. It is easy to see that applying Theorem 2.3 to eah onseutive pair of matries

C

i

wehavethat

C

1

,

C

2

,...,

C

p

−

1

arenonsingular

M

-matries. Thisanbe shownbyindution.

WenowextendTheorem 2.10tothesub-diretsum of

p

submatriesof agiven nonsingular

M

-matrix

M

. Tothatend,werstdene

M

(

S

)

aprinipalsubmatrixof

M

withrowsandolumnswithindiesinthesetofindies

S

=

{

i, i

+ 1

, i

+ 2

, . . . , j

}

. In[2℄wealltheseonseutiveprinipalsubmatries. Forexample,matries

A

and

B

givenby(2.12)anbeexpressedasasubmatriesof

M

givenby(2.11)as

A

=

M

(

S

1)

,

(9)

Theorem 2.13. Let

M

be a nonsingular

M

-matrix. Let

A

i

=

M

(

S

i

)

,

i

=

1

, . . . , p

,beprinipal onseutivesubmatriesof

M

andonsiderthe

p

−

1

k

i

-subdiret

sumsgiven by

C

i

=

C

i

−

1

⊕

k

i

A

i

+1

,

i

= 1

, . . . , p

−

1

,

(2.20)

inwhih

C

0

=

A

1

. Theneahofthe

k

i

-subdiretsums

C

i

isanonsingular

M

-matrix. Proof. It is easy to relate the struture of eah

C

i

to that of the submatries

A

i

involved. Weonsiderthat

A

i

areoverlappingprinipal submatriesoftheform

(2.12)butallowingthateah

A

i

hasdierentnumberofbloks. Let

M

bepartitioned as

M

=







M

11

M

12

M

13

· · ·

M

1

n

M

21

M

22

M

23

· · ·

M

2

n

M

31

M

32

M

33

· · ·

M

3

n

. . .

. .

. . . .

M

n

1

M

n

2

M

n

3

· · ·

M

nn







(2.21)

aordingwith the size of theprinipal submatries

A

i

. Eah blok

M

ij

may bea

submatrixofmorethanone

A

m

,

m

= 1

, . . . , p

. Let

b

(

l

)

ij

≥

0

bethenumberofmatries

A

m

suhthat

M

ij

isasubmatrixof

A

m

,for

m

= 1

, . . . , l

+ 1

. Ofourseweanhave

b

(

ij

l

)

= 0

. Letusonsiderthe

l

thsubdiretsum

C

l

,

1

≤

l

≤

p

−

1

,whihisoftheform

C

l

=







b

11

(

l

)

M

11

b

(

12

l

)

M

12

b

13

(

l

)

M

13

· · ·

b

(

1

l

)

M

1

l

b

(

21

l

)

M

21

b

(

l

)

22

M

22

b

(

l

)

23

M

23

· · ·

b

(

l

)

2

l

M

2

l

b

(

31

l

)

M

31

b

(

l

)

32

M

32

b

(

l

)

33

M

33

· · ·

b

(

l

)

3

l

M

3

l

. . .

. .

. . . .

b

(

l

1

l

)

M

l

1

b

(

l

2

l

)

M

l

2

b

(

l

3

l

)

M

l

3

· · ·

b

(

ll

l

)

M

ll







.

(2.22)

Observe that

C

l

is a

Z

-matrix and that

b

(

l

)

ii

>

0

. Furthermore, for eah olumn it

holdsthat

b

(

l

)

ii

≥

b

(

l

)

ji

, j

= 1

, . . . , l

.

The proofproeeds in amanner similar to that of Theorem 2.10. Considerthe

Z

-matrix(partitionedinthesamemanneras

M

)

T

l

=

M

l

diag(

b

(

l

)

11

I, b

(

l

)

22

I, b

(

l

)

33

I, . . . , b

(

l

)

ll

I

)

,

where

M

l

istheprinipalsubmatrixof(2.21)withrowandolumnbloksfrom

1

to

l

. It followsthat

T

−

1

l

≥

O

and therefore

T

l

isanonsingular

M

-matrix. Finally, sine

C

l

≥

T

l

,weonludethat

C

l

isanonsingular

M

-matrix,

l

= 1

, . . . , p

.

Example 2.14. Given the nonsingular

M

-matrix

M

of Example 2.11, let us onsiderthefollowingoverlappingbloks

A

1

=

M

(

{

1

,

2

,

3

}

) =





13

/

14

−

4

/

23

−

3

/

20

−

3

/

7

21

/

23

−

1

/

5

−

1

/

7

−

7

/

46

17

/

20



(10)

A

2

=

M

(

{

2

,

3

,

4

,

5

}

) =







21

/

23

−

1

/

5

−

1

/

21

−

1

/

93

−

7

/

46

17

/

20

−

1

/

14

−

1

/

186

−

27

/

92

−

1

/

15

4

/

7

−

58

/

93

−

9

/

46

−

3

/

10

−

1

/

7

53

/

62







,

A

3

=

M

(

{

3

,

4

,

5

,

6

}

) =







17

/

20

−

1

/

14

−

1

/

186

−

2

/

23

−

1

/

15

4

/

7

−

58

/

93

−

27

/

92

−

3

/

10

−

1

/

7

53

/

62

−

9

/

46

−

2

/

15

−

2

/

7

−

7

/

62

83

/

92







.

Thenwehavethe

2

-subdiret sum

C

1

=

A

1

⊕

2

A

2

=







13

/

14

−

4

/

23

−

3

/

20

0

−

3

/

7

42

/

23

−

2

/

5

−

1

/

21

−

1

/

93

−

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







,

whih isanonsingular

M

-matrix,andthe

3

-subdiretsum

C

2

=

C

1

⊕

3

A

3

=







13

/

14

−

4

/

23

−

3

/

20

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

−

2

/

15

−

2

/

7

−

7

/

62

83

/

92







,

whihisalsoanonsingular

M

-matrixinaordanewithTheorem2.13. Observethat in this example we have

k

1

= 2

and

k

2

= 3

. Note also that, for example, wehave

b

(1)

22

= 2

,

b

(1)

33

= 2

,

b

(1)

14

= 0

,

b

(2)

22

= 2

,

b

(3)

22

= 2

,

b

(2)

33

= 3

,

b

(2)

14

= 0

.

3. Subdiret sumsof inverses. Let

A

and

B

be nonsingularmatries parti-tionedasin (2.1). Inthis setionweonsider the

k

-subdiret sumof theirinverses. We will establish ounterparts to some of results in the previous setions. Let us denoteby

G

=

A

−

1

⊕

k

B

−

1

,with

A

−

1

and

B

−

1

partitionedasin(2.3),i.e.,

G

=





ˆ

A

11

A

ˆ

12

0

ˆ

A

21

A

ˆ

22

+ ˆ

B

11

B

ˆ

12

0

B

ˆ

21

B

ˆ

22





.

(3.1)

As aorollaryto, andin analogyto Theorem 2.1, the nextstatementindiates thatthenonsingularityof

A

22

+

B

11

isaneessaryonditiontoobtain

G

nonsingular. Theorem 3.1. Let

A

and

B

benonsingular matriespartitionedasin(2.1)and let their inverses be partitioned as in (2.3). Let

G

=

A

−

1

(11)

We remark that in analogyto the expression (2.5) of

C

−

1

, the expliit form of

G

−

1

is

G

−

1

=





A

11

−

A

12

H

−

1

A

21

A

12

−

A

12

H

−

1

A

22

A

12

H

−

1

B

12

B

11

H

−

1

A

21

B

11

H

−

1

A

22

−

B

11

H

−

1

B

12

+

B

12

B

21

H

−

1

A

21

B

21

H

−

1

A

22

−

B

21

H

−

1

B

12

+

B

22





.

(3.2)

Corollary3.2. When

A

and

B

arenonsingular

M

-matrieswith the ommon blok

A

22

=

B

11

asquarematrixof order

k

,i.e., ofthe form

A

=

A

11

A

12

A

21

A

22

,

B

=

A

22

B

12

B

21

B

22

,

(3.3)

then

H

= 2

A

22

isnonsingularandtherefore

G

=

A

−

1

⊕

k

B

−

1

isnonsingular.

We note that this isthe asewhen

A

and

B

are overlappingsubmatriesof an

M

-matrix,i.e.,oftheform(2.12)and(2.11)onsideredinSetion2.2,wherewewere

interestedinthesubdiretsumof

A

and

B

. Hereweonludethatthesubdiretsum oftheirinversesisalwaysnonsingular.

Example3.3. Let

A

and

B

bethematriesofExample2.11,thenaordingto Corollary3.2,the

3

-subdiretsumoftheinverses

G

=

A

−

1

⊕

3

B

−

1

≈







1

.

5033 0

.

5513 0

.

5547 0

.

2757 0

.

3912

0

.

9540 1

.

5996 0

.

7158 0

.

3635 0

.

4038

0

.

6004 0

.

5636 2

.

9750 0

.

8144 0

.

7407 0

.

3708

2

.

0383 2

.

1242 3

.

5729 6

.

5498 5

.

3372 2

.

0139

0

.

8953 0

.

9650 2

.

0470 1

.

8025 3

.

9062 0

.

9048

0

.

8551 1

.

3803 1

.

2652 1

.

9143







isanonsingularmatrix.

Intheaboveexampleadiretomputationshowsthat

G

−

1

isnotan

M

-matrix:

G

−

1

≈







0

.

8900

−

0

.

2337

−

0

.

0750

−

0

.

0119

−

0

.

0511

0

.

0512

−

.

0054

0

.

0470

−

0

.

0714

−

1500

−

0

.

0714

0

.

4274

−

0

.

0978

.

0565

0

.

7123







whih is nota

Z

-matrix. Note that when

A

and

B

are

M

-matries we havefrom (3.1) that

G

=

A

−

1

⊕

B

−

1

is nonnegative. Therefore assuming that

G

−

1

exists we have

(

G

−

1

)

−

1

≥

O

. Thenitisanaturalquestiontoseekonditionssothat

G

−

1

isa nonsingular

M

-matrix. Westudythisquestionnext.

Theexpressions (3.1)of

G

and (3.2)of

G

−

1

, Theorem3.1, and theobservation thatfornonsingular

M

-matrieswehave

(

G

−

1

)

−

1

≥

O

, implythefollowingresult. Theorem 3.4. Let

A

and

B

benonsingular

M

-matriespartitioned asin (2.1) andtheir inverses partitioned asin (2.3). Let

G

=

A

−

1

⊕

k

B

−

1

with

k

≥

1

, andlet

H

=

A

22

+

B

11

benonsingular. Then

G

−

1

isa nonsingular

M

-matrixif andonly if

G

−

1

(12)

Corollary 3.5. Let

A

and

B

be lower and upper bloktriangular nonsingular

M

-matries, respetively, partitioned asin (2.10) with

A

22

and

B

11

square matries

of order

k

and

H

=

A

22

+

B

11

nonsingular. Then

G

−

1

= (

A

−

1

⊕

k

B

−

1

)

−

1

is a

nonsingular

M

-matrixifandonly ifthe followingonditionshold: i)

B

11

H

−

1

A

21

≤

O

. ii)

B

11

H

−

1

A

22

isa

Z

-matrix.

iii)

−

B

11

H

−

1

B

12

+

B

12

≤

O

.

Proof. >From(3.2)and(2.10)wehavethat

G

−

1

=





A

11

0

B

11

H

−

1

A

21

B

11

H

−

1

A

22

−

B

11

H

−

1

B

12

+

B

12

0

B

22





(3.4)

andtherefore

G

−

1

isa

Z

-matrixifandonlyiftheonditionsi),ii)andiii) hold. Conditionsi),ii)andiii)intheorollaryarenotasstringentastheymayappear. Forexample,let

A

and

B

bebloktriangularnonsingular

M

-matriespartitionedas in(2.10)with aommonblok

A

22

=

B

11

,asquarematrixoforder

k

, i.e.,

A

=

A

11

0

A

21

A

22

and

B

=

A

22

B

12

0

B

22

.

(3.5)

Then

G

−

1

= (

A

−

1

⊕

k

B

−

1

)

−

1

is anonsingular

M

-matrix,sine we havefrom (3.4)

that

G

−

1

=





A

11

O

1

2

A

21

1

2

A

22

1

2

B

12

O

B

22





,

andtherefore

G

−

1

isa

Z

-matrix. Infat,inthisase,wehave

G

=





A

−

1

11

O

−

A

−

1

22

A

21

A

−

1

11

2

A

−

1

22

−

A

−

1

22

B

12

B

−

1

22

O

B

−

1

22





≥

O.

Thenextexampleillustratesthissituation.

Example3.6. Let

A

and

B

bethematriesofExample2.8,then

G

=

A

−

1

⊕

2

B

−

1

=







1

/

3

0

1

/

8

49

/

80 21

/

80

9

/

20

7

/

24 77

/

80 73

/

80 11

/

10

0

1

/

2







,

and

G

−

1

=







3

0

−

18

/

49 146

/

49

−

6

/

7

−

39

/

49

−

4

/

7

−

22

/

7

2

−

11

/

7

0

2



(13)

isanonsingular

M

-matrixinaordanewithCorollary3.5.

Note that if the hypotheses of Corollary 3.5 are satised, and realling Theo-rem2.6,wehavethat eahofthematries

C

=

A

⊕

k

B

and

G

−

1

= (

A

−

1

⊕

k

B

−

1

)

−

1

arebothnonsingular

M

-matries.

4.

P

-matries. A square matrix is a

P

-matrix if all its prinipal minors are positive. As aonsequene we havethat all the diagonal entries of a

P

-matrix are positive. Itisalsofollowsthatanonsingular

M

-matrixisa

P

-matrix. Itanalsobe shownthat if

A

isanonsingular

M

-matrix,then

A

−

1

isa

P

-matrix;see,e.g.,[5℄. In [3℄ it is shown that the

k

-subdiret sum (with

k >

1

) of two

P

-matries is not neessarilya

P

-matrix. Our results in Setions 2.1 and 3 hold for nonsingular

M

-matriesandinversesof

M

-matries,respetively. Asthesetwolassesofmatries

are subsetsof

P

-matries, it is naturalto ask ifsimilar suient onditionsan be foundsothatthe

k

-subdiretsumof

P

-matriesisa

P

-matrix. Thefollowingexample indiates thattheanswermaynotbeeasytoobtain,sineevenin thesimplestase ofdiagonalsubmatriesthe

k

-subdiretsummaynotbea

P

-matrix.

Example 4.1. Giventhe

P

-matries

A

=





543 388 322

69

160

0

368

0

375





,

B

=





136

0

219

0

225 159

61

177 230





wehavethatthe

2

-subdiretsum

C

=

A

⊕

2

B

=







543 388 322

0

69

296

0

219

368

0

600 159

0

61

177 230







isnota

P

-matrix,sine

det(

C

)

<

0

.

Aknowledgment. Wethanktherefereeforaveryarefulreadingofthemanusript andforhisomments.

REFERENCES

[1℄ A.BermanandR.J.Plemmons.NonnegativeMatriesintheMathematialSienes,Aademi Press,NewYork,1979.Reprintedandupdated,SIAM,Philadelphia,1994.

[2℄ R.Bru, F.Pedrohe,andD.B.Szyld.AdditiveShwarzIterationsforMarkovChains,SIAM J.MatrixAnal.Appl.,toappear.

[3℄ S.M.FallatandC.R.Johnson.Sub-diretsumsandpositivitylassesofmatries.LinearAlgebra Appl.,288:149173,1999.

[4℄ A. Frommer and D.B. Szyld. Weighted Max Norms, Splittings,and Overlapping Additive ShwarzIterations,NumerisheMathematik,83:259278,1999.

[5℄ R.A.HornandC.R.Johnson.MatrixAnalysis.CambridgeUniversityPress,NewYork,1985. [6℄ B.F.Smith,P.E.Bjørstad,andW.D.Gropp.Domaindeomposition: Parallelmultilevel meth-odsforelliptipartialdierentialequations.CambridgeUniversityPress,Cambridge,1996. [7℄ A.ToselliandO.B.Widlund.DomainDeomposition:AlgorithmsandTheory.SpringerSeries

inComputationalMathematis,vol.34. Springer,NewYork,2005.