2.4 Irreducible Matrices

Reducible

nMA is reducible

if there is a permutation P such that

22

1211

0 A

AAAPPT

where A11 and A22 are square matrices

each of size at least one; otherwise

A is called irreducible.

1 x 1 matrix: irr or reducible

11a

By definition,

every 1 x 1 matrix is irreducible.

Some authors refer to

irreducible if a≠0

reducible if a=0

digraph

nij MaA

nV ,,2,1

Let

the digraph of A is the digraph with

0;, ijajiE

denoted by G(A).

Example for diagraph

00

1 ALet

G(A) is

1

2

strongly connected

A digraph is called strongly connected

if any vertices x,y, there is a directed path

from x to y , and vice versa.

Remark 2.4.1

nnjiMA n ,,2,1,,

Z

Let

Given

0ijAIf ,then there is a directed

walk in G(A) of length l from vertex i to vertex j.

If A is nonnegative, then converse also holds

.

,,,,,,

0

..,,,1

,0

322

32

,,,1

322

32

32

2

jvertextoivertexfrom

lengthofwalkdirectedaistherehence

Ejiiiii

aaa

tsniii

thenAIf

aaaA

jiiiii

ij

jiiiniii

iiij

An Equivalent relation on V

Define a relation ~ on V by i~j if i=j or

i≠j and there is a directed walk from

vertex i to vertex j and vice versa.

~ is an equivalent relation.

Strongly Connected Component

The strongly connected components are

precisely the subgraphs induced by vertices

that belong to a equivalent class.

How many strongly connected components are there ?

see next page

There are five strongly connected components.

final strongly connected component

final strongly connected component

Theorem 2.4.2

nMALet

The following conditions are equivalent:

InjandIiaij \0

(a) A is irreducible.

(b) There does not exist a nonempty proper

subset I of <n> such that

(c) The graph G(A) is strongly connected.

.

,0

0

,,1,,1

)(

:

,,,,

\0

..

)(~)(~)()(

221122

1211

)()(

11

reducibleisAHence

MAandMAwhereA

AA

formtheofisAPPHence

aaAPP

kjandnkifor

andSthen

nmim

bynnDefine

iiIandiiILet

InjandIiifa

tsnIthatSuppose

abbyba

knk

T

iijiijT

n

m

nkkc

ij

ji

0)(,),1(

,)(,),1(

0

,,1,,1

11

0

.

,

)(~)(~)()(

)()(

1111

22

1211

cIIc

jiijT

knk

T

AandkI

thennkILet

aPAP

kjandnkiforthen

nksomefor

MAandMAwhere

A

AAPAP

tsPnpermutatioaisthere

thenreducibleisASuppose

babyab

cij

c

IjandIiathen

IjandIiEji

tsnIthen

componentsconnectedstrongly

moreortwohasAGThen

connectedstronglynotisAGthatSuppose

bcBycb

0

),(

..

.

)(

.)(

)(~)(~)()(

.)(

)(

,0

..

)(~)(~)()(

connectedstronglynotisAGHence

IyandIxyvertextoxvertexfrom

AGinpathnoistherethatclearisit

thenA

tsnIthatSuppose

abbybc

c

II c

Exercise 2.4.3 p.1

2212

11 0

AA

AAPPT

(a) Show that a square matrix A is reducible

if and only if there exists a permutation

such that

where A11,A22 are square matrices each of

size at least one.

kkn

T

T

mmjiijT

cji

i

nkkc

k

II

MAandMAwhere

AA

AAPP

formtheofisAPPthen

aaAPP

ImjandImi

nkkj

andkiforThen

nimi

bynnDefine

mmmI

andmmmILet

AtsnIThmby

thenreducibleisAIf

ji

c

2211

2221

11

)()(

21

21

0

0

)()(

,,2,1

,,2,1

)(

:

,,,

,,,

0..,1.4.2

,""

.,1.4.2

0)(,),1(

,)(,),1(

0

,,1,,1

0

""

)()(

2211

2212

11

reducibleisAThmBy

AandnkI

thenkILet

aAPP

nkjandkiforthen

MAandMAwhere

AA

AAPP

formtheofisAPPthatsuch

PnpermutatioaexiststhereIf

cIIc

jiijT

knk

T

T

Exercise 2.4.3 p.2

(b) Deduce that if A is reducible, then so is AT

.

0

.

0

..

)(,

22

1211

2211

2212

11

reducibleisAthen

DefinitionBy

A

AAPAPPAPthen

oneleastatsizeofeach

matricessquareareAandAwhere

AA

APAP

tsPnpermutatioexiststhere

abythenreducibleisAIf

T

T

TTTTTT

T

Theorem 2.4.40, AMA n

Let

The following conditions are equivalent:

0)( 1 nIA

(a) A is irreducible.

(b) A has no eigenvector which is

semipositive but not positive.i.e.

every semipositive eigenvector of A is

positive

(c)

.

0

0,0,0

0

00

,

,

00

..

)(

..,00

))(~)(~(:1

)()(

22

1211

21121

121

11

2221

1211

2221

1211

11

reducibleisAHence

B

BBAPPthen

BhavemustsoxBBut

xB

xx

BB

BB

thenBB

BBAPPTake

xPxAPPPSince

Rxwherex

xP

tsNkandPnpermutatiothen

AsomeforxAx

tspositivenotisxxthatSuppose

abbyMethod

ba

T

T

TTT

kT

.,2.4.2

,0

0

0

,0;

)(

..,00

))(~)(~(:2

1

reducibleisAThmBy

IiIja

Iixaxa

IixAx

thenxiILet

AsomeforxAx

tspositivenotisxxthatSuppose

abbyMethod

cij

jIj

ijj

n

iij

ii

i

c

.

0

0)(

0

)(

000

),(

0

.

)(~)(~

)()(

111111

22

1211

11

11

22

1211

positivenotbutvesemipositiiswhich

Aofreigenvectoanisy

then

yA

yAyAy

A

AA

thenAtoingcorrespond

RyreigenvectovesemipositiahasA

ThmFrobeniusPerronBy

A

AA

formtheofisAthatassumemayWe

generalityofloseWithout

reducibleisAthatSuppose

bathatshowtogoingareWe

ab

k

.

)(

1

)(

,1

0)(..1,1

0)(

0)(

0

1

1

2

1

1

1)(

)()(

1

1

121

eirreduciblisA

connectedstronglyisAG

nmostatlenghtof

jvertextoivertexfromAGin

walkdirectedaistherenji

Atsnknji

IA

Hence

niIA

AFor

An

nA

nA

nIIA

ca

ijk

n

iin

nn

positivenotbut

IA

AIAIA

A

AA

formtheofisAthatassumemayWe

generalityofloseWithout

reducibleisAthatSuppose

cathatshowtogoingareWe

acofproofAnother

nn

,0

0

0

.

)(~)(~

)()(

1

22

12111

22

1211

Remark

nMA

If the degree of minimal polynomial of

,2,1,0;

,,,, 12

jAspan

AAAIspanj

m

is m,then

,1,

,,,,

,,,

,,,

,,,

)(0

,)(

1

1

11

1

1110

1110

1110

mmkfor

AAIspanAyInductivel

AAIspan

AAAspanA

thatfollowsIt

AAIspanA

AaAaIaA

AAaAaIatmthen

ttataatmLet

mk

m

mmm

mm

mm

m

mmmA

mmmA

Theorem 2.4.5

0, AMA nLet

0)( 1 mIA

A is irreducible if and only if

where m is the degree of minimal polynomial

of A.

.

.)(

)(

,2,10)(

,,1,,,,

1,,10)(

0)(..,1

""

.4.4.2

0)(

,0)("''

1

1

1

1

reducibleisAthen

connectedstronglynotisAGthen

jtoifromAGinwalkdirectedanotistherethen

kA

mmlAAIspanASince

mkAthen

IAtsnji

contrarythetoAssume

eirreduciblisAThmby

IAthen

nmIAthatAssume

ijk

ml

ijk

ijm

n

m

Exercise 2.4.6

0, AMA nLet

0 someforxAx

A is irreducible if and only if

for any semipositive vector x, if

then x>0

0

,2.4.2

,0

0

,00

0

,0;

.

0

..0""

1

xHence

ioncontradictiswhich

reducibleisAThmby

IjIia

Iixaxa

xandASince

IiAx

thenxiILet

positivenotisxthatSuppose

someforxAx

tsRxLet

cij

Ijjijj

n

jij

i

i

n

c

.,4.4.2

0,0

,""

eirreduciblisAThmby

positivebemust

AofreigenvectovesemipositieveryHence

xthensomeforxAxif

xvectorvesemipositianyforthatAssume

Theorem 2.2.1 p.1

(Perron’s Thm)thenAIf ,0

0)( A

)()( AA (b)

(c)

(a)

uAAutsu )(..0

eigenvaluesimpleaisA)(

)(),()( AAA

(f)

(g)

(e)

1,)(,)(

,)(

lim

vuandvAvAuAAu

whereuvA

A

TT

Tm

m

(d)

A has no nonnegative eigenvector

other than (multiples of) u.

Theorem 2.3.5 (Perron-Frobenius Thm)

XAAX )(

0A

..00 tsX

)()( AA , thenIf

and

Corollary 2.4.72,0, nAMA nLet

If A is irreducible, then the conclusions

(a),(b),(c),(d) and (f) of Perron Thm all hold.

.0)(

.

0,0,0

0,0)(

04.4.2,

)(..,00)()(

,

)(),(),(Pr

AHence

Aoflityirreducibithescontradictwhich

AsoxABut

AxthenAIf

xThmbyeirreduciblisASince

xAAxtsxandAA

matrixenonnegativforThmFrobiniusPerronBy

candbaforoof

.1

.

).(

,

.

..

.,

..)(

.

.1)(

.)()(

isAofmultiplegeometrictheHence

AoflityirreducibithescontradictThis

Atoingcorrespond

AofreigenvectoanisyxClearly

positivenotbut

vesemipositiisyxtsRthen

tindependenlinearlyisyx

tsAtoingcorrespondRyreigenvecto

anotherhasAThennotSuppose

isAofmultiplegeometricthe

thatshowtoFirst

eigenvaluesimpleisAthatshowTod

n

).(max)(

)()(

)())(()(

)()()(

))(()(

,0)(

)(,0

.0

.0arg

,)))(((

))((..

.

.2)(

11

11

11

1

AAthatfactthescontradictwhich

AA

smallsufficientisif

yIAAyIAA

yIAAyIAA

yIAAAyIA

IASince

yAAyxSince

ythatassumemayweHence

xythenelsufficientIf

RxxyAASince

xyAAtsRyThen

notSuppose

orderofAtoingcorrespondreigenvecto

generatingnohasAthatshowtoremainsIt

nn

nn

nn

n

n

uofmultiplethanother

reigenvectoenonnegativnohasAHence

impossibleiswhich

XvceA

XvA

XvAAXvXv

vAAvthen

vAvAtsvLet

Athen

uAAuanduwhere

uofmultipleanotisXandXAX

tsXthatSuppose

fofoof

T

T

TTT

TT

T

0sin,)(

0))((

)(

)(

)(..0

)(

)(0

..00

)(Pr

Exercisse 2.4.8nRyu ,Let

Prove that if u is positive and y is nonzero

then there is a unique real scalar c such that

u+cy is semipositive but not positive.

0;max

,0,

0

,,,10

,,10:1

0

0

0

0

0

ii

i

i

i

ii

i

i

i

i

yy

x

nisomefory

x

andyify

x

nisomeforyx

andniyx

positivenotbutvesemipositiisyx

niyIfCase

0;min

0;max

,0,

,0,

0

,,,10

0:2

0

0

0

0

0

ii

i

ii

i

i

i

ii

i

ii

i

i

i

i

yy

xor

yy

x

nisomefory

x

andyify

x

yify

x

nisomeforyx

andniyx

positivenotbutvesemipositiisyx

nisomeforyIfCase

Remark 2.4.9

xAAx )( , then x must be

If A is nonnegative irreducible matrix and

if x is nonzero nonnegative vector such that

the Perron vector of A.

.

)(

)(max)(

)()(

)()()(

..0

0)(

)()()(

0)()(

0)(

4.4.2,

0)(0

,)(

11

1

11

1

1

Aofvectorperrontheisxthen

xAAxHence

AAthatfactthescontradictwhich

AA

xIAAxIAA

ts

xIAwhere

xIAAxIAA

xAAxIAthen

IA

ThmbyeirreduciblisASince

xAAx

thenxAAxthatSuppose

nn

n

nn

n

n

Exercise 2.4.10

(n,1) is irreducible.

Show that the nxn permutation matrix with

1’s in positions (1,2), (2,3), …, (n-1,n) and

.,4.4.2

0)(

0)(1,:1

0)(1,:1

0)(

01

1

2

1

1

1)(

1

1

1

1

121

eirreduciblisPThmBy

PIHence

PIPthenjiIfCase

PIPthenjiIfCase

njianyFor

niPI

Pn

nP

nP

nIPI

n

ijn

ijjin

ijn

ijij

iin

nn

Exercise 2.4.11 (a)

(a) If A is irreducdible, then AT is

irreducdible

In below A and B denote arbitrary nxn

nonnegative matrices. Prove or disprove

the following statements:

eirreduciblisA

AI

AI

AI

eirreduciblisA

doI

T

nT

Tn

n

0)(

0)(

0)(

)(

1

1

1

eirreduciblisA

arcsitsof

directionthegreverby

AGfromobtainedisAG

connectedstronglyisAG

connectedstronglyisAG

eirreduciblisA

doTeacher

T

T

T

)

sin

)()((

)(

)(

)(

Exercise 2.4.11 (b)

(b) If A is irreducible and p is a positive

integer, then Ap is irreducible.

.10

01

)(

,01

10

.

)(

2 reducibleisABut

eirreduciblisA

connectedstronglyisAG

thenALet

examplecounterGivenNo

doTeacher

.

0100

1000

0001

0010

0001

0010

1000

0100

0001

0010

1000

0100

.

,

0001

0010

1000

0100

2

reducibleis

ABut

eirreduciblisA

thenALet

Exercise 2.4.11 (c)

(c) If Ap is irreducible for some positive

integer, then A is irreducible.

.

0

*

0

,

0

.

.

)(

22

11

22

1211

1211

22

1211

reducibleisAthen

A

A

A

AAAPPPAP

haveweZnanyFor

oneleastatsizeofmatricessquareareAandAwhere

A

AAAPPthatsuch

Pmatrixnpermutatioaistherethen

reducibleisAthatSuppose

Yes

doTeacher

n

n

n

nnTnT

T

.

.)(

)(

)(0

..1

1

0

,1

eirreduciblisA

connectedstronglyisAG

AGinjvertex

toivertexfromwalkdirectedaisthere

AA

tsnk

njianyforthen

AI

theneirreduciblisAIf

ijpk

ij

kp

np

p

Exercise 2.4.11 (d)

(c) If A and B are irreducible, then

A+B is irreducible.

.)(

,)(),(

).(

)()(.

)()(),(:Pr

.),(

""

0,:2

.0

,

""

,0,:1

)(

connectedstronglyisBAGthen

connectedstronglyisBGAGofoneIf

BAGofsubgraph

bothareBGandAGsetvertexsame

thehaveBAGandBGAGoof

eirreduciblisBAofonethatassumeonlyneed

Yesisanswerthethen

BAthatassumeweIfCase

reducibleisBAthen

BAifexampleFor

Noisanswerthethen

BAassumenotdoweIfCase

doTeacher

.

.

.

00

,00

0)(

..

.

22

1211

22

1211

22

1211

eirreduciblisBAHence

scontradictwhich

reducibleareBandA

B

BBBPPand

A

AAAPP

BPPandAPPSince

C

CCBPPAPPPBAP

tsPnpermutatioThen

reducibleisBAthatSuppose

TT

TT

TTT

Exercise 2.4.11 (e)

(e) If A and B are irreducible, then

AB is irreducible.

.10

01

,01

10,

)(

reducibleisABBut

eirreduciblareBandAthen

BAletexampleFor

doTeacher

.

1000

0100

0010

0001

0010

0001

0100

1000

0001

0010

1000

0100

.

0010

0001

0100

1000

0001

0010

1000

0100

reducibleis

ABBut

eirreduciblareBandAthen

BandALet

Exercise 2.4.11 (f)

(f) If all eigenvalues of A are 0,

then A is reducible.

0:

)(

,,

)(

AException

AofeigenvalueanisA

theneirreduciblisAIfYes

doTeacher

.

0)(,7.4.2

.

0)(,0

reducibleisAHence

scontradictwhich

ACorollaryBy

eirreduciblisAthatSuppose

AthenareAofseigenvalueallIf

Exercise 2.4.11(g)

001

001

110

The matrix is reducible.

G(A) is strongly connected, then

A is irreducible

1

2

3

Exercise 2.4.11(h)

0011

0001

1100

0100

The matrix is reducible.

G(A) is not strongly connected, then

A is reducible.

1

2

3

4

0100

0000

0101

1010

1010

1000

0101

0100

0001

0100

0010

1000

0001

0100

0010

1000

0011

0001

1100

0100

0001

0100

0010

1000

0001

0010

1000

0100

,

0001

0100

0010

1000

PP

thenPLet

T

Exercise 2.4.11(i)

0001

0010

1000

0100

The matrix is reducible.

G(A) is strongly connected, then

A is irreducible.

1

2

3

4

Exercisse 2.4.11(j)

AIn

A is irreducible if and only if

is irreducible.

.

.

)()(.

)(

graphaofsconnectnesstrongthe

affectnotdoes

loopsofabsenceorpresencetheBut

loops

morehavemayformerthethatexceptsamethe

areAGandIAGdigraphsTheYes

doTeacher

.

.

0

0)(

..

.""

22

1211

22

1211

eirreduciblisAIHence

ioncontradictiswhich

reducibleisA

IB

BIBAPP

B

BBPAIP

tsPnpermutatiothen

reducibleisAIthatSuppose

T

T

.

.)(

)(

,0)(.1

1

1

12

1

12

)2(0

.

""

121

1

eirreduciblisATherefore

connectedstronglyisAGHence

AGinjvertextoivertexfrom

walkdirectedaisthere

thenAtsnk

njianyFor

An

nA

nI

AIThen

eirreduciblisAIthatAssume

ijk

nnn

n

eirreduciblisAI

AI

AIeirreduciblisA

proofgAlternatin

n

n

0)2(

0)(

:

1

1

Exercisse 2.4.11(k)

If AB is irreducible, then BA is irreducible

.01

0101

1

111

1

1

.11

1111

1

101

1

1

,111

101

1

1

.

)(

reducibleisBAbut

eirreduciblisABthen

BandALet

No

doTeacher

.

000

111

111

001

001

110

000

101

011

.

011

011

101

000

101

011

001

001

110

000

101

011

001

001

110

reducibleisBABut

eirreduciblis

ABthen

BandALet

Exercise 2.4.12.

)(min)()(max)(11

ARAorARA ini

ini

Let A be an nxn irreducible nonnegative

matrix. Prove that if

then all row sums of A are equal. Give

an example to show that the result no longer

holds if the irreducibility assumption is removed .

.

00

0,)(

0)(,0

0)(

)(

)(

)(..0

),(min

.

),(min)(

)(

1

1

sumsrowequalhasAHence

reAethen

reAeanduBut

reAeuthenrAIf

rAeuSince

eurA

eruAeueuA

uAAu

uAuAtsuLet

reAethenARrLet

sumsrowequalhasA

thenARAifthatshowTo

doTeacher

T

T

T

T

TTT

TT

T

ini

ini

.

)()(,4.3.2

0

)())()((

)())((

0)()(

0)(,

)(

),(max)(:1

11.

.

11

11

1

1

1

equalareAofsumsrowallHence

impossibleiswhichAAExerciseBy

smallsufficientfor

eAIAeAIA

eAIAeAIA

AeeAAI

AIeirreduciblisASince

vesemipositiisAeeA

thenARAIfCase

ReLetnotSuppose

equalareAofsumsrowallthatshowTo

nn

nn

n

n

ini

nT

.

),()(

0

)())()((

)())((

0)()(

0)(,

)(

),(min)(:2

11

11

1

1

1

equalareAofsumsrowallHence

impossibleiswhichAA

smallsufficientfor

eAIAeAIA

eAIAeAIA

eAAeAI

AIeirreduciblisASince

vesemipositiiseAAe

thenARAIfCase

nn

nn

n

n

ini

.

)(

9.4.2Re

,

)(

,)(min)(

:2

1

equalareAofsumsrowallthethen

eAAe

markby

eirreduciblisASince

eAAe

thenARAIf

CaseforproofgAlternatin

ini