2.4 Irreducible Matrices
Dec 31, 2015
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