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2.5 Matrix With Cyclic Structure. Remark. When A is an irreducible nonnegative matrix. # of distinct peripheral eigenvalues of A. = the index of imprimitivity of G(A). = cyclic index of A. = spectral index of A. = the largest m such that. sgn(τ). A cyclic permutation. transpositions. - PowerPoint PPT Presentation
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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 p
AAApk )()()()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
00
00
0
00
aa
aa
aaa
aa
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
aaaaaaaaaaaa
aaaa
aaaaaaaaA
Example 2.5.2
000
000
00
00
43
31
2421
1411
a
a
aa
aa
A
G(A)
1 2
34
taaatattcthen
aaAE
AE
aaaaaaAE
circuitsdisjobyered
becannotGofverticestheceAthen
circuitandloopthe
namelycircuitstwopreciselyhasAG
A 4314313
114
111111
1
2
43143143143113
3
)(
1)(
0)(
1)(
intcov
sin,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
aaaaac
andaaaaaaaaaacThen
aaaaaa
aaaaaaaaaaa
aaaaaa
aaaaaa
aaaaaaA
cacaAsIf
Csr
k
jj
j AA1
1 )(1)(1
k ,,, 21
src
of the form
where αis a path from vertex r to vervex s
In general,
k
j
kn
j
AA1
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
tctcHence
nBA
nDBDA
BDDA
BA
Fact
circuiteachforBABGAGBAD
)()(),()(~
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
010
001
100
,
001
100
010
BA
3,detdet BA
1)1(det 21321313 aaaB
1)1(det 31231213 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
TDD
BAorBA ~~?
ko.
A and B have the same principal minors
Hartfiel and Loewy proved the following:
.4
3
.,min
~~)(
nfornotbut
nforsufficientisconditionThe
eirreduciblisAthenorsprincipal
ingcorrespondsamethehaveBandAwhenever
BAorBAsatisfiesFMAIf TDD
n
Introduce A Semiring 0; aRaR
abba
baba ,max
are associative,commutative
R+ form a semiring under
On
and
introduce
by:
distributes over
0 is zero element
and
and
max-product algebra A B ⊕ p.1
nmnppm MBAMBMA ,
410
023
010
,
112
100
013
BA
kjikk
ij baBA max
kjkk
j xaxA max
A B ⊕ p.2
423
410
033
BA
412
123
013
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
k
iijiiiiiij
k 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 length
two 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.