2.5 Matrix With Cyclic Structure

Remark

)()(2

AAe mi

# of distinct peripheral eigenvalues of A

= cyclic index of A

When A is an irreducible nonnegative matrix

= the index of imprimitivity of G(A)

= spectral index of A

= the largest m such that

sgn(τ) nk Siii 21

kiiiiii 13121

11)sgn( k

A cyclic permutation

a product of 1 transpositions.

sgn(π)

nk S 21

i

)sgn()sgn()sgn( 1 k

Any permutation can be writen of the form:

where each is cyclic permuation.

)()2(2)1(1)sgn(det nnS

aaaAn

k

j

kn

k

j

k

kk

j

j

k

k

A

A

AA

aaa

1

1

11

1

)()2(2)1(1

)()1(

)()1()1(

)()()sgn()sgn(

)sgn(

1

1

n

j

n

j

k

n

k

n

S

k

j

kn

S

k

j

Sk

Skk

A

A

AA

aaaA

1

1

11

1

)()2(2)1(1

)()1(

)()1()1(

)()()sgn()sgn(

)sgn(det

1

1

Remark

0det AIf

is n.

If G(A) has no circuits, then detA=0

, then there are (vertex-)

disjoint circuits in G(A), the sum of lengths

Ek(A)

p

jj

j A1

1 )(1

pkp ,,,1 1

)(AEk = the sum of nonzero terms of the

(or pAAApk

)()()()1(21

form

the sum of lengths is equal to k.

where

)

are disjoint circuits

Remark

))(0( nA AAc

nkAEk ,,1,0)(

If G(A) has no circuits, then

nn

k

knk

knA ttAEttc

1)()1()(then

i.e. A is nilpotent

Example 2.5.1

4443

3432

232221

1211

0000

000

aaaa

aaaaa

A

G(A)1 2

4 3

)()1()(

)()1(

)()1()(

)1()()1(

)()1())(()1()(

))()(()1(

))()(()1())(()1(det

44221111

1

44224411221122

43343223211212

2

44221133

3443221123

3223441123

44211223

3

322344113443221143342112

3223441134

3443221134

2112344324

aaaAE

aaaaaa

aaaaaaAE

aaaaaaa

aaaaaaaAE

aaaaaaaaaaaaaaaa

aaaaaaaaA

Example 2.5.2

000000

0000

43

31

2421

1411

aa

aaaa

AG(A)1 2

34

taaatattcthen

aaAE

AEaaaaaaAE

circuitsdisjobyeredbecannotGofverticestheceAthen

circuitandloopthenamelycircuitstwopreciselyhasAG

A 4314313

114

111111

1

2

43143143143113

3

)(

1)(

0)(1)(

intcovsin,0det

)3,4(),4,1(),1,3(31:,,)(

Cofactor of A

ts

02211 tnsntsts cacaca

Aofcofactorrscsr ),(Let

snsnssss cacacaA 2211det

If

n

jtjsj

n

jjtsjst

Tij

caadjAaadjAA

thentsif

cadjAandIAadjAASince

110

,

,)(det

Example stcConsider

1 2

3

4

by digraph

5

6

554462263113

5543346126554433612612

554462263113

5543346126554433612612

55446226311346

55433461261236

55443361261246

13131212

,)(

)()1(

)()1(

)1(det

det,1

aaaaacandaaaaaaaaaacThen

aaaaaaaaaaaaaaaaa

aaaaaa

aaaaaa

aaaaaaA

cacaAsIf

Csr

k

jj

j AA1

1 )(1)(1

k ,,, 21

srcof the form

where αis a path from vertex r to vervex s

In general,

k

j

knjAA

1

1 )()(1

is the sum of possible terms

are circuits s.t. the path andand

or

the circuits are mutually disjoint and together they contain all vertices of G(A)

Dn

nD

13221)1( iiiiii k

aaa

nD complete digraph of order n

circuit 121: iiii k

Consider

as an edge-weighted digraph

weight of

weight of D(π)

)(D

nS

permutation digraph

product of weights of circuits))(( Dwt

Example

4)5(,5)4(,1)3(,3)2(,2)1(5 andS

))(())(( 5445312312 aaaaaDwt

Let

1

23

4

5

then

Remark 2.5.5 (i)

nMBA ,

)()( tctc BA

If A and B have the same set of circuits

for each circuit

Let

)()( BA

and as a consequence

and

then A and B have equal corresponding

principal minors of all possible orders

)()(detdet

1

1

tctcHencenBA

nDBDA

BDDA

BA

Fact circuiteachforBABGAGBA

D )()(),()(~

eirreduciblA:

cf. Exercise 2.4.20

why does A must be irreducible? see next second page.

1

1

1 1

1 1

1

2

G(A)=G(B) is not irreducibole

But A and B are not diagonally similar.

)()( BA

does not appear in circuit

product

Remark 2.5.5 (ii)

BAD~

minors

A and B have equal corresponding principal

A and B have the same

circuit products.

principal minors

A and B have the same corresponding

A Counter example

Counter Example 1

010001100

,001100010BA

3,detdet BA1)1(det 213213

13 aaaB1)1(det 312312

13 aaaA

G(A):

1 2

3

G(B):

1 2

3

1

111

1

1

Counter Example 2

)()( BGAG

TT AA )(

A and AT have the same principal minors

But we may have

Question

impossible

TDDBAorBA ~~

?ko.

A and B have the same principal minors

Hartfiel and Loewy proved the following:

.43

.,min

~~)(

nfornotbutnforsufficientisconditionThe

eirreduciblisAthenorsprincipalingcorrespondsamethehaveBandAwhenever

BAorBAsatisfiesFMAIf TDD

n

Introduce A Semiring 0; aRaR

abba

baba ,max

are associative,commutative

R+ form a semiring under

On

andintroduce

by:

distributes over

0 is zero element

and

and

max-product algebra A B ⊕ p.1

nmnppm MBAMBMA ,

410023010

,112100013BA

kjikkij baBA max

kjkkj xaxA max

A B ⊕ p.2

423410033

BA

412123013

BA

Fuzzy Matrix Version 1,0, ijij aaA

,minmeet

1,0,

,maxjoin

max-min algebra

ijijij baBA

kjik

p

kij

nm

nppm

baBA

MBA

MBMA

1)(

,

Boolean Matrix

positive1

elementzero0 ,min

001,111,010,000

B: (0,1)-matrix

111,110,000

,max

different from F2

Spectial case of max-min algebra

0nnA

timesk

k AAAA

In max-product algebra, max-min algebra

satisfies the associative low.

)()( AGAG kk

in the sence max algebra or fuzzy algebra

2k

n

jiij aaA1

2

02ijA a directed walk of length

two in G(A) from i to j

ijkA

jiiii k 121

11

1211,, k

kii

jiiiiiijk aaaA

0ijkA a directed walk of length

k in G(A) from i to j

Furthermore,

G(A) contain the directed walk

the sum of walk products of A

w.r.t the directed walks in G(A) from i to j of length k

In the setting of Max algebra (max-product algebra) p.1

ijAA

0A

0ijAA a directed walk of lengthtwo in G(A) from i to j

the maximum of walk products of

A w.r.t the directed walks in G(A) from i to j of length two

jiij aaAA max

In the setting of Max algebra (max-product algebra) p.2

ijk A

0ijk A a directed walk of length k

in G(A) from i to j

the maximum of walk products of

A w.r.t the directed walks in G(A) from i to j of length k

In the setting of Fuzzy Matrix (max-min algebra)

ijk A

0A jiij aaAA

,minmax

0ijk A a directed walk of length k

in G(A) from i to j

It is difficult to explain the geometric

meaning of

In the setting of Fuzzy Matrix (max-min algebra) p.2

ijAA

0A jiij aaAA

,minmax

0ijk A a directed walk of length k

in G(A) from i to j

Furthermore, G(A) contain the directed walk

It is difficult to explain the geometric

meaning of

from i to j of length k

Combinatorial Spectral Theory of Nonnegative Matrices

Plus max alge yxyx ,max

yx

yx

eeyx

eeyx

)exp(

)exp(

yxyx

RR:exp

They are isometric

Remark p.2

,0

XxxTxT )()(

FXT :space and

Then for any

Let X be a Topology space, F is a Banach

FXT :

is continuous

map such that T(X) is precompact in F.

there is a continuous

map of finite rank s.t.

sgn(π)

nk S 21

)sgn()sgn()sgn( 1 k

11)sgn(

i

A permutation

where each

k

j

kn

k

j

k

kk

j

j

k

k

A

A

AA

aaa

1

1

11

1

)()2(2)1(1

)()1(

)()1()1(

)()()sgn()sgn(

)sgn(

1

1

is cyclic permuation.