Chapter 2

Nonnegative Matrices

2-1

Introduction

Entrywise nonnegative

,0nmA

(entrywise ) nonnengative means

jiaij ,0 different from positive semidefinite

Strictly positive

0nmA

strictly positive means

jiaij ,0

different from positive definite

Remark

000 AA

e.g.

01

10

nonzero, nonnegative but not positive

semipositive≡nonzero, nonnegative

BA

0 BA

jiba ijij ,

00,0 ABBA

CBCABCACCBA ,0,

000,0 AAxandxA

000,0 AABandBA

Remark

000 BorAAB

e.g.

00

10BA

2-2

Perron’s Theorem

spectral radius

)(CMA n

spectral radius

)(max)( AA

譜半徑

Example

30

01 iA

3)( A

ijnm aA

ijnmaA

BAAB

Proven in next page

BAABHence

BA

ba

ba

ABAB

ij

n

kkjik

n

kkjik

ijij

1

1

Collatz Wielandt

0A

collatz weilandt

XAXtsXA ..000)(

Lemma 2.2.2 (1)

)()( AA

0A

Proven in next page

)(

0

A

AzA

zA

Az

zz

Lemma 2.2.2 (2)

)(A

0A

Proven in next page ( 證明很重要 )

is closed and bounded above

.)(

)(

,

01,lim

1,..00

,

,lim)(

:)(

)(

)(0

closedisAHence

Awthen

wXAX

klettingbyobtainwe

XwAXSince

XandXthenXXLet

XesubsequencconvergentahasX

XXwAXtsX

Nkfor

thenwwandAwIf

AofCloseness

A

A

kkk

k

k

iii

ik

ii

kkkkk

kkNkk

.)(

)(

.max

00

)(

,111

:)(

aboveboundedisAHence

Aforboundupperanis

xe

xe

xe

xAew

eAethen

AofsumcolumnimunthebeLet

wxe

xAe

xewwxeAxe

xsomeforwxAx

AwanyFor

ReLet

aboveboundedisA

TT

TT

TT

TT

TTT

nT

Remark

andAAA )()(max)(0

0A

Proven in next page

uAAutsu )(..0

)(00

0

0,00

)(

.0)()(

)(

0)(

0

,

..00

),(max

max

max

max

max

max

max

max

max

max

max

max

max

Aandu

Auu

AuuandASince

uAu

A

smallsuffiAuAuA

AuAuA

uAuA

ABut

vesemipositiisuAu

thenuAuSuppose

uAuthatshowTo

uAutsu

thenALet

generalized eigenvector

NkAsomefor ),(

u is called generalized eigenvector ofA if

0)( uIA k

Remark

AJAPP 1

Proven in next page

the columns of P are the

generalized eigenvectors of A.

11

23323

12212

111

21

21

1

)(

)(

)(

0)(

*0

01

01

kkkkk

nk

nk

A

A

PPIAPPAP

PPIAPPAP

PPIAPPAP

PIAPAP

PPPP

PPPPA

PJAP

JAPP

tocorrklenghofchainJordancalled

PPP

PIAPIAPIAP

and

PIA

PIA

PIA

PIA

kk

kk

kkk

kk

.

)(,,)(,)(,

0)(

0)(

0)(

0)(

121

12

33

22

1

Remark

eigenvaluegleisA sin)(

The geometric multiple of λ =1 and

there is no generalized eigenvector

other than eigenvector corr. to λ

Remark

.sin)( eigenvaluegleisA

0A

Proven in next page

1)(

)()()(0

..

,,0

)(,)(..,

.

1)(1

AofmultiplegeometricHence

vuAvuABut

positivenotbutvesemipositiisvu

tsRchoosecanwethen

tindependenlineararevuandu

vAAvuAAutsRvu

notSuppose

AofmultiplegeometricthatshowoT

n

.sin)(

)(max)(,

)()(

.0))((

)0()(

)(

0

,.sin.

)(

)())((..

,

)(.

2

eigenvaluegleisATherefore

AAthatbecauseimpossibleiswhich

AA

smallsuffallforyAAy

uyAAy

uyAAy

ythatassumemaywe

smallsuffrgchooByRranyfor

ruybyyreplacemaywethatNote

uyAAy

uAAuwhereuyIAAtsRy

thennotSuppose

Atorrecorseigenvectothanother

reigenvectodgeneralizenohasAthatshowTo

n

Remark

)()( AandA

0A

Proven in next page

)()( AandA

)(

argarg

,,1

)(.

)(

,)(

)(,)(

)(

)(

..0

1

1111

A

othereachofmultiplearezandz

zz

zazazaza

nkfor

Atocorresp

AofreigenvectoaniszandzAAz

zAAzzzA

thenAIf

AthenAifthatshowTo

A

A

zAAzz

zAztsCz

n

nknknknk

n

Remark

If A>0, then A has no nonnegative

eigenvector other than

(multiple of) u , where u>0 and

uAAu )( Proven in next page( 證明很特

別 )

uofmultiplethanother

reigenvectoenonnegativnohasAHence

impossibleiswhich

XvceA

XvA

XvAAXvXv

vAAvthen

vAvAtsvLet

Athen

uAAuanduwhere

uofmultipleanotisXandXAX

tsXthatSuppose

T

T

TTT

TT

T

0sin,)(

0))((

)(

)(

)(..0

)(

)(0

..00

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.

Norm on a vector space

Vxx 0

(i)

(iii)

(ii) scalarxx

is a norm on V

= hold iff x=0

Vyxyxyx ,

d

we introduce a metric

is a metric space

yxyxd ),(

with

on V, by

dV ,

kasxxk

kasxxd k 0,

Convergent matrix sequence

LA Nkk

NkMA nk

can be interpreted in

where

ijk

ijk

lanjiji

)(lim,1,

one of the following equivalent way:

(i)

ijkijk lLaA ,)(

kasLAk 0

is in any fixed norm of

where

nM

The topology of

(ii)

nM is independent of

(the maximum norm)

to be

ijnjiaA

,1max

we obtain (i)

In (ii), take

Bounded matrix sequence

NkkA

,2,1,1..0 )( knjiMatsM kij

(ii)

(i)

,2,1..0 kMAtsM k

is bounded means

Fact 2.2.4

kk

kk

kkk

BABA

limlimlim

)lim)(lim(lim kk

kk

kkk

BABA

(ii)

(i)

kp

k

k

k

k

k

k

p

k

A

A

A

A

A

A

lim

lim

lim

lim

2

1

2

1

(iii)

Apply of Fact 2.2.4 (ii)

PgularnonsomeforJAPP A sin1

LAkk

lim

PAPPAP kk

kk

)lim(lim 11

and P is nonsigularIf

then

convergent problem of A is corresponding to convergent problem of

AJ

Theorem 2.2.3

)(1 A 1)( A

nMA

NkkA

Let

(i) The sequence

converges to the zero matrix iff

11

1)( A

1)( A

NkkA

(ii) converges iff

or

and 1 is the only eigenvalue

with modulus 1 and the corresp.

Jordan blocks are all

)(A

1)( A

1)( A

NkkA

(iii) is bounded iff

either

and

if then

or

1 1)( Av

Lemma 2.2.5

1)(lim

km

kJ

1

1

0)(lim

km

kJ (i) If

(ii) If

then

and m=1, then

1 convergesk

1

1

Nkk

mJ )(

(iii) If

the sequence

and m=1, then

is bounded

Note:

In this case, the seqence does not

converge if

explain in next page

θ

2m

1

1

Nkk

mJ )(

(iv) If

then the sequence

or

is unbounded

and

)0(

0

1

0

010

1

01

)(

)(

1

00miifNN

i

kN

i

k

NwhereNI

J

i

iim

i

ikik

i

ik

k

k

km

0)(lim

0!

)1()1(

01

2

2

12

)(

1

2

1

21

121

km

k

ikik

k

k

kk

k

kk

kkk

mkkkk

km

J

kasi

ikkki

k

and

k

kk

kk

m

kkk

J

unboundedisJ

kaskk

mandCase

unboundedisJCase

k

kk

kk

m

kkk

J

iv

Nkk

m

k

Nkk

m

k

kk

k

kk

kkk

mkkkk

km

)(

21:2

)(1:1

2

2

12

)(

)(

1

1

2

1

21

121

Exercise 2.2.7

)(),( AA

nMA

)(A

)0)(( A

eigenvalue and

is non-nipotent

for every

Suppose that )(A is a simple

Exercise 2.2.7k

k AA

)(lim

what can you tell about the vector

Prove that

x and y?

exists and is of the form *xy

2-3

Nonnegative Matrices

Lemma 2.2.2

)(A0A

is closed , bounded above and

If , then

)()(: AA

Lemma 2.3.1

0AIf , then

)()(max AA

XXA

AX

A

A

XXA

A

XAAX

tsXthen

AAtsSuppose

AAshowtoremainsIt

AAhavealreadyWe

0

2

0

0

0

00

)()(

)(

)(

..00

)()(..0

0)()(

)()(

00,

0

0)(

lim

3.2.2

1)(

)(

)(

,2,1)(

,

0

00

0

Xasoncontraditiaiswhich

X

A

A

TheoremBy

A

A

A

ABut

kXXA

A

yInductivel

k

k

k

Lemma 2.3.2

BA0If , then

)()( BA

)()(

)()(max)(

)()(

)(

)(..00

)()(

BA

BBA

BA

BXXA

BXAXso

BABut

AXXAtsX

AA

Fact

)()()( 2121 AAAA

)(),(max)( 2121 AAAA

Corollary 2.3.3

0, AMA n

)()( AB

, and B is a principal submatrix of A

If

then

In particular )(max1

Aaniii

)()(

)()ˆ()(

,00

0ˆ

**

*

AB

ABB

thenAB

BLet

BAthatassummay

generalityofloseWithout

Exercise 2.3.4

BAMBA n 0,,

)()( BA then

If

Hint: There is some α>1 such that

BAA 0

)()(

),()()()(0

)()()(

,2.3.2

0

,1,1,0;min

,10

0)(:2

)()(0

,0)(:1

0)(,

0,0

BAhaveweso

BAAA

BAA

LemmaBy

BAA

thennjiaa

bLet

njibaSince

AthatAssumeCase

BA

thenAthatAssumeCase

BhaveweThmPerronby

BBASince

ijij

ij

ijij

Theorem 2.3.5 (Perron-Frobenius Thm)

XAAX )(

0A

..00 tsX

)()( AA , thenIf

and

)(#)()(lim

)()()()(

0

,1

,2,1

321

321

existsAA

AAAA

AAAAAlso

keachforAthen

jiak

AandMALet

kFor

kk

k

ijijknk

)(),#(#)(#

)#(#)(

)(

,

)(1,00

,

1,,1,11)(

..0

,

AandBy

Ahence

A

xAxkLetting

xAxAandxexthen

xx

sayesubsequencconvergentahasIt

sequenceboundedaisx

ewherexeandxAxA

tsx

ThmPerronbykeachFor

kkkk

k

iiiiT

i

Nkk

Tk

Tkkkk

k

Ri (A)

n

jija

1

)(ARi = i th row sum of A

Cj (A)

n

iija

1

)(AC j = j th column sum of A

Corollary 2.3.6

)(max)()(min11

ARAAR ini

ini

)(max)()(min11

ACAAC jnj

jnj

0 AandMA n

Then

Let

and

)(max)()(min

)(

,0

)(

)(

)(..00

Re

,111

),(max,)(min

)(max)()(min)1(

11

11

11

ARAAR

RAr

havewesoezBut

eRzezAerz

eRzAezerz

zAAzthen

zAzAtszisthere

ThmFrobeniusPerronBy

Aere

thenRe

andARRARrLet

ARAARthatshowTo

ini

ini

T

TTT

TTT

TT

T

nT

ini

ini

ini

ini

)(max)()(min

)()(),()(

)(max)()(min

)(max)()(min

),2(

)(max)()(min)2(

11

11

11

11

ACAAC

AAtctcSince

ACAAC

ARAAR

haveweBy

ACAACthatshowTo

jnj

jnj

TAA

jnj

Tj

nj

Tj

nj

TTj

nj

jnj

jnj

T

Matrix norm

nM

)()()( BNANABN

is called a matrix norm if N( - ) is a

A norm N( - ) on

norm on

, and N( - ) is submultiplicative i.e.

nM

Matrix norm Induced by Vector norm

nC

nM

AXAx 1max

be a (vector) norm on

Let

Define on by

matrix norm induced by the vector norm

Proposition of matrix norm induced by vector norm

BABA

BAAB

BAx

ABxAB

xBABxABxAABx

xAAx

x

AxAxA

BA

BxAx

BxAx

xBABA

x

xx

xx

x

x

0

01

11

1

1

max

)(

maxmax

maxmax

max

)(max

Remark 2.3.7

nMA

nM

)(AA

is a marix norm on If

then

)(

..0)(

AA

A

AXAXXA

XAX

tsXchooseandALet

not Euclidean

matrix normcorrect proof in

next page( 很重要 )

)(

..0)(

AAHence

A

BBABBAthen

BXXX

AXAXXA

XXXAAB

MXXXBLet

XAX

tsXchooseandALet

n

Special norm:l∞,lp

llp p

ini

n

l

1

2

1

max

pn

i

p

i

n

pl

1

1

2

1

Special Matrix norm

nM

nCl

be the matrix norm on

Let

induced by the norm of

Corollary

0, AMA n

)(A

If the row sums of A are constant

Let

then A row sum of A

)(),#(#)(#

)#(#)(

max

1

1

1

:

:

)(#)(

)(

1

111

1

21

ArandBy

Arso

AAxrHence

eandrAeBut

raXaxa

nisomeforxaAX

XwithCxxxXFor

pf

ArClaim

Arr

Ar

reAethen

AofsumrowcommontheberLet

x

n

jij

n

jij

n

jjij

n

jjij

nn

Exercise 2.3.8 p.1

1

A

n

iij

njaA

111max

max absolute column sum of A

Exercise 2.3.8 p.2

A

n

jij

niaA11

max

max absolute row sum of A

n

iij

njx

n

iij

nj

n

iijjj

n

iij

n

iij

nj

n

iij

nj

n

iij

nj

n

jj

n

iij

nj

j

n

j

n

iij

njj

n

j

n

iij

n

i

n

jjij

n

i

n

jjij

n

ii

nTn

n

iij

nj

aAxAHence

aaAAethen

njsomeforaaLet

a

xaxa

xaxa

xaxaAxAx

xwithRxxxxanyFor

aAthatshowTo

11111

11111

0111

11

111111

1 111 1

1 11 111

121

111

maxmax

max

1max

max

maxmax

max

1

max)1(

1

000

0

n

jij

nix

n

jij

ni

n

jjii

n

jji

n

jiji

nTniii

n

jji

n

jij

ni

n

jij

ni

n

jij

ni

n

jjij

ni

n

jjij

nii

ni

nTn

n

iij

ni

aAxAHence

aAy

thenaAyand

iaaAyandythen

RaaayLet

aatsniLet

axa

xaxaAxAx

xwithRxxxxanyFor

aAthatshowTo

111

11

1

11

21

1110

1111

11111

21

11

maxmax

max

,

1

)sgn()sgn()sgn(

max..

maxmax

maxmaxmax

1

max)2(

00

0

000

0

Exercise 2.3.9

)(A

0, AMA n

Prove that if A has a positive eigenvector, then the corresponding eigenvalue is

Let

[Hint: Apply the Perron-Frobenius Thm

to AT ]

)(#)(

0,000

)(#)(

)(

)(

)(

)(..00

..)(

,0

byA

uvvanduSince

uvAuv

uvAuv

uvAAuv

vAAv

vAvAtsv

ThmFrobeniusPerronBy

uAutsA

thenAofeigenvalueanbeuLet

T

TT

TT

TT

TT

T

Remark 2.3.10

1)( AA

AA)(

0, AMA n

If A has equal row sums, then

Let

If A has equal column sums, then

AAHence

AA

CorollaryBy

ArrA

Ar

ExercisebyArandreAethen

sumrowcommontheberLet

solutioneAlternativ

ArAExerciseBy

AofreigenvectopositiveiseSince

ExercisebyArandreAethen

sumrowcommontheberLet

AAthatshowTo

)(

)(

,6.3.2

)(

)(

8.3.2,

.

:

.)(9.3.2

,

8.3.2,

.

)()1(

1

1

1

1

1

1

1

)(

)(,6.3.2

)(

)(

8.3.2

.

:

.)()(9.3.2

,

8.3.2

.

)()1(

AAHence

AACorollaryBy

ArrA

Ar

ExercisebyArandreeAthen

sumrowcommontheberLet

solutioneAlternativ

ArAAExerciseBy

AofreigenvectopositiveiseSince

ExercisebyArandreeAthen

sumcolumncommontheberLet

AAthatshowTo

T

T

T

a row stochastic matrix

0, AMA n

with row sums all equal to 1,then

A is called a row stochastic matrix.

If

1)( AandeAe

a column stochastic matrix

0, AMA n

with column sums all equal to 1,then

A is called a column stochastic matrix.

If

1)( AandeAe TT

Exercise 2.3.11

)(min1

ARini

)(A

AB

))(

,,)(

,)(

(21 ARARAR

diagn

[ Hint: Let

Deduce Corollary 2.3.6 from Remark

2.3.9 and Lemma 2.3.2

To show that inequality

consider B=DA, where D is the diagonal

matrix

show that

))(

,,)(

,)(

(

0:2

)(0:1

0),(min

)()(min)1(

21

1

1

ARARARdiagDwhere

DABConsider

Case

ACase

thenARLet

AARthatshowTo

n

ini

ini

)(),#(#)(#

)#(#)(,9.3.2

,,,2,1)(

)(#)()(

0

,,,2,11)(

0

)(

)(

)(

22

11

Aandby

BExerciseby

eeBandniBRBut

AB

AB

niAR

Since

AAR

AAR

AAR

Bthen

i

i

nn

0)(

0)(0

,0)(0

0)()(

),,,(

0),(max

)(max)()1(

21

1

1

ARif

ARifc

andARif

ARifARd

withddddiagDwhere

CDABConsider

thenARLet

ARAthatshowTo

i

ii

i

iii

n

ini

ini

)(max)(

)(),#(#)(#

)#(#)(,9.3.2

,,,2,1)(

)(#)()(

0

0)(1)(

100)(

1ARAHence

Aandby

BExerciseby

eeBandniBRBut

AB

ABthen

ARifAR

andnjaARSince

ini

i

ii

iji

Diagonally Similar p.1

ADDB 1

00,0 BDA

00,0 BDA

nMBA , are diagonal similar

In particular

if there is nonsingular matrix D s.t.

Diagonally Similar p.2

00,0 BDA

00,0 BDA

preserves the class of nonnegative

(as well as , positive) matrices.

In particular

nonnegative diagonal similarity

Corollary 2.3.12

nCx

n

i i

ijj

nj

n

i i

ijj

nj x

axA

x

ax

1111max)(min

n

jjij

ini

n

jjij

inixa

xAxa

x 1111

1max)(

1min

0, AMA nThen for any positive vector

and

we have

n

j i

ijj

nj

n

j i

ijj

nj

n

jjij

ini

n

jjij

ini

n

jjij

ini

n

jjij

ini

jiji

n

nTn

x

axA

x

ax

haveweSimilarlly

xax

Axax

xax

ADDxax

CorollaryBy

xax

ADDofentryji

thenxxxdiagD

letandRxxxxLet

1111

1111

11

1

11

1

21

21

maxmin

,

1max

1min

1max

1min

6.3.2

1),(

,),,,(

0),,,(

Exercise 2.3.13 p.1

nTn Rxxxx ,,, 21

For any semipositive vector

Wielandt numbers of A with respect to x are defined and denoted respectively by:

the upper and the lower Collatz-

0, AMA nLet

Exercise 2.3.13 p.2

xAxxRA :0inf)(

wxAxwxrA :0sup)(

(we adopt the convention that inf ψ=∞)

Prove that for any semipositive x, we have

Exercise 2.3.13

0:)(

max)( ii

iA x

x

AxxR

0:min)( ii

iA x

x

Axxr

0;max)(

max

0..

0..

,,2,1

0

ii

iA

i

i

x

ii

i

iii

ii

xx

AxxRHence

x

Ax

xtsix

Ax

xtsixAx

nixAx

xAx

i

0;min)(

min

0..

0..

,,2,1

0

ii

iA

i

i

x

ii

i

iii

ii

xx

AxxrHence

x

Ax

xtsix

Ax

xtsixAx

nixAx

xAx

i

Exercise 2.3.14 p.1

)(A

Axxts ..0

0, AMA n

(i) Prove that if

Let

for some positive vector x then

)(

min)(

,12.3.2

min

0sin,,,1

,,1

1

1

AHence

x

AxA

haveweCorollaryBy

x

Ax

xcenix

Ax

nixAx

Axx

i

i

ni

i

i

ni

i

i

ii

Exercise 2.3.14 p.2

)(A

xAxts ..0

0, AMA n

(ii) Prove that if

Let

for some positive vector x then

)(

max)(

,12.3.2

max

0sin,,,1

,,1

1

1

AHence

x

AxA

haveweCorollaryBy

x

Ax

xcenix

Ax

nixAx

xAx

i

i

ni

i

i

ni

i

i

ii

Exercise 2.3.14 p.3

)(A

0, AMA n

(iii) Use parts (i) and (ii) to deduce that

Let

if A has a positive eigenvector thenthe corresponding eigenvalue is

)(

)()(

),()(

0,00

)(

,

AHence

AandA

haveweiiandiBy

uAuanduAu

uandASince

AsomeforuAu

thenAofreigenvectopositiveabeuLet

Exercise 2.3.15 p.1

2221

1211

aa

aa

2221

1211

aa

aa

are diagonally similar.

Show that the matrices

and

.

10

01

10

01

2221

1211

2221

1211

2221

1211

2221

1211

similardiagonallyare

aa

aaand

aa

aathen

aa

aa

aa

aa

Exercise 2.3.15 p.2

333231

232221

131211

aaa

aaa

aaa

are diagonally similar ?

Are the matrices

and

333231

232221

131211

aaa

aaa

aaa

.

1,,

1,1,1

..),,(

sin

.

323121

31

231

121

1

332321

31311

3

3231

2221211

2

3131

12121

111

3

2

1

333231

232221

131211

13

12

11

333231

232221

131211

321

similardiagonally

notarematricestwothisHence

impossibleiswhich

ddanddddd

ddanddddd

adaddad

dadadad

daddada

d

d

d

aaa

aaa

aaa

d

d

d

aaa

aaa

aaa

tsddddiagD

matrixdiagonalgularnonisthereThen

matricesdiagonalarematricestwotheseSuppose