Tropical Linear Algebra: Orbits and the Core of a Matrixweb.mat.bham.ac.uk/P.Butkovic/files/Beamer Eindhoven.pdf · 2013. 3. 8. · Peter Butkoviµc (joint work with H. Schneider

Tropical Linear Algebra:Orbits and the Core of a Matrix

Peter Butkoviµc (joint work with H. Schneider and S. Sergeev)

Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013

1.Tropical linear algebra


Max-times, max-plus and nonnegative linear algebra

Ground set: R+ = fx 2 R; x � 0g

Two semirings:

(R+,+, .)

(R+,�, .) def= (R+,max, .)

(R+,max, .) ' (R[ f�∞g ,max,+) via x �! log x


Max-times, max-plus and nonnegative linear algebra

Ground set: R+ = fx 2 R; x � 0gTwo semirings:

(R+,+, .)

(R+,�, .) def= (R+,max, .)

(R+,max, .) ' (R[ f�∞g ,max,+) via x �! log x


Passages between max and lin

S.Friedland (1984):

A ... an irreducible nonnegative matrixnAko∞k=1

... sequence of Hadamard powers of A

λk ... the Perron root of Ak

Then kp

λk �! λ(A) (in max-times)

V.Maslov+V.Kolokoltsov (1980s): "dequantisation"

limh�!0+

h. log�ea/h + eb/h

�= max (a, b)



S.Friedland (1984):

A ... an irreducible nonnegative matrix

nAko∞k=1



Then kp



limh�!0+


�= max (a, b)



S.Friedland (1984):

A ... an irreducible nonnegative matrixnAko∞k=1



Then kp



limh�!0+


�= max (a, b)


Tropicalisation

Line segment

xy = fαx + βy ; α, β � 0, α+ β = 1g

Tropical line segment

xy = fαx � βy ; α, β � 0, α� β = 1g


Extension to matrices and vectors

A� B = (aij � bij )

A B =�∑�k aik bkj

�


Extension to matrices and vectors

A� B = (aij � bij )A B =

�∑�k aik bkj

�


Some basic properties

Compared to (R+,+, .) in (R+,max, .) we are

losing invertibility of �gaining idempotency of �

A�1 exists () A is a generalised permutation matrix

[good for cryptography!]

Idempotency: a� a = a

(a� b)k = ak � bk , if k � 0(A� B)k may or may not be Ak � Bk

(I � A)k = I � A� A2 � ...� Ak




losing invertibility of �

gaining idempotency of �





(I � A)k = I � A� A2 � ...� Ak









(I � A)k = I � A� A2 � ...� Ak





A�1 exists () A is a generalised permutation matrix[good for cryptography!]



(I � A)k = I � A� A2 � ...� Ak







(a� b)k = ak � bk , if k � 0

(A� B)k may or may not be Ak � Bk

(I � A)k = I � A� A2 � ...� Ak








(I � A)k = I � A� A2 � ...� Ak


Basic problems

One-sided max-linear systems (fast algorithms exist):

A x = b

A x � bA x = λxA x � λx


Basic problems


A x = bA x � b

A x = λxA x � λx


Basic problems


A x = bA x � bA x = λx

A x � λx


Basic problems


A x = bA x � bA x = λxA x � λx


Basic problems

Two-sided max-linear systems (no polynomial method isknown):

A x = B x

A x = B yA x � c = B x � dA x = λB x (generalized eigenproblem)


Basic problems


A x = B xA x = B y

A x � c = B x � dA x = λB x (generalized eigenproblem)


Basic problems


A x = B xA x = B yA x � c = B x � d

A x = λB x (generalized eigenproblem)


Basic problems


A x = B xA x = B yA x � c = B x � dA x = λB x (generalized eigenproblem)


Basic problems

Max-linear programming:

f T x �! min (max)s.t.A x = b

f T x �! min (max)s.t.A x � c = B x � d


Basic problems

Periodicity of matrix powers:A,A2,A3, ...

Periodicity of matrix orbits:A x ,A2 x ,A3 x , ...Tropical characteristic polynomial and equationLinear independence, regularity, rank,...


Basic problems


Periodicity of matrix orbits:A x ,A2 x ,A3 x , ...

Tropical characteristic polynomial and equationLinear independence, regularity, rank,...


Basic problems


Periodicity of matrix orbits:A x ,A2 x ,A3 x , ...Tropical characteristic polynomial and equation

Linear independence, regularity, rank,...


Basic problems


Periodicity of matrix orbits:A x ,A2 x ,A3 x , ...Tropical characteristic polynomial and equationLinear independence, regularity, rank,...


Permanent:per(A) = ∑

π∏iai ,π(i )

Tropical permanent:

maper(A)df= ∑

π

�∏i

ai ,π(i )

(max,+)= max

π∑iai ,π(i )

Hencemaper(A) = ∑

iai ,π(i )

for at least one πThe permanent is called strong if = for exactly one π



π∏iai ,π(i )

Tropical permanent:

maper(A)df= ∑

π

�∏i

ai ,π(i )

(max,+)= max

π∑iai ,π(i )

Hencemaper(A) = ∑

iai ,π(i )

for at least one π

The permanent is called strong if = for exactly one π



π∏iai ,π(i )

Tropical permanent:

maper(A)df= ∑

π

�∏i

ai ,π(i )

(max,+)= max

π∑iai ,π(i )

Hencemaper(A) = ∑

iai ,π(i )

for at least one πThe permanent is called strong if = for exactly one π


Linear dependence/independence

Columns of A 2 Rm�n+ are called

LI ifAk = ∑�j 6=k αj Aj

is not true for any k and αj 2 R+Gondran-Minoux LI if

∑�j2U αj Aj = ∑�j2V αj Aj

is not true for any U,V � f1, ..., ng ,U \ V = ∅,U,V 6= ∅and αj 2 R+strongly LI if A x = b has a unique solution for someb 2 Rm

Strong LI =) Gondran-Minoux LI =) LIStrong regularity =) Gondran-Minoux regularity =)regularity

Strong rank � GM rank � Rank



Columns of A 2 Rm�n+ are calledLI if

Ak = ∑�j 6=k αj Ajis not true for any k and αj 2 R+

Gondran-Minoux LI if








Ak = ∑�j 6=k αj Ajis not true for any k and αj 2 R+Gondran-Minoux LI if


is not true for any U,V � f1, ..., ng ,U \ V = ∅,U,V 6= ∅and αj 2 R+

strongly LI if A x = b has a unique solution for someb 2 Rm









Strong LI =) Gondran-Minoux LI =) LI

Strong regularity =) Gondran-Minoux regularity =)regularity









Strong rank � GM rank � RankPeter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013

A 2 Rn�n+ is

Gondran-Minoux regular if and only if maper (A) is attainedby permutations of only one parity [GM 1979]

(This is equivalent to the EVEN CYCLE problem [PB 1991]... O

�n3�)

strongly regular if and only if maper (A) is attained by onlyone permutation [PB 1983] ... O

�n3�

Other rank concepts: Barvinok rank, Kapranov rank


A 2 Rn�n+ isGondran-Minoux regular if and only if maper (A) is attainedby permutations of only one parity [GM 1979]

(This is equivalent to the EVEN CYCLE problem [PB 1991]... O

�n3�)

strongly regular if and only if maper (A) is attained by onlyone permutation [PB 1983] ... O

�n3�

Other rank concepts: Barvinok rank, Kapranov rank


Brief overview of tropical spectral theory

A x = λx ... tropical eigenproblem

V (A,λ) = fx 2 Rn+;A x = λ xg ,λ 2 R+Λ(A) = fλ > 0;V (A,λ) 6= f0gg



A x = λx ... tropical eigenproblemV (A,λ) = fx 2 Rn+;A x = λ xg ,λ 2 R+

Λ(A) = fλ > 0;V (A,λ) 6= f0gg



A x = λx ... tropical eigenproblemV (A,λ) = fx 2 Rn+;A x = λ xg ,λ 2 R+Λ(A) = fλ > 0;V (A,λ) 6= f0gg


Brief overview of tropical spectral theory: maximum cyclemean

Given A 2 Rn�n+ the (geometric) mean of a cycleγ = (i1, ..., ik ):

µ(γ,A) = kpai1 i2ai2 i3 ...aik i1

Maximum cycle mean of A :

λ(A) = max fµ(γ,A);γ cycleg

If µ(γ,A) = λ(A) ... γ is a critical cycleIndices on critical cycles are critical indices







If µ(γ,A) = λ(A) ... γ is a critical cycle

Indices on critical cycles are critical indices







If µ(γ,A) = λ(A) ... γ is a critical cycleIndices on critical cycles are critical indices


Brief overview of tropical spectral theory: the Kleene star

A� = I � A� A2 � ...

A� = I � A� A2 � ...� An�1 if and only if λ (A) � 1λ�(λ (A))�1 A

�= 1



A� = I � A� A2 � ...A� = I � A� A2 � ...� An�1 if and only if λ (A) � 1

λ�(λ (A))�1 A

�= 1



A� = I � A� A2 � ...A� = I � A� A2 � ...� An�1 if and only if λ (A) � 1λ�(λ (A))�1 A

�= 1


Brief overview of tropical spectral theory: the principaleigenvalue

(Cuninghame-Green, 1962) For any A, λ(A) is

an eigenvalue of Athe greatest (principal) eigenvalue of Athe only eigenvalue of A whose corresponding eigenvectorsmay be positivethe unique eigenvalue if A is irreducible (in this case alleigenvectors are positive)

Every eigenvalue of A is the maximum geometric cycle meanof some principal submatrix

Columns of�(λ (A))�1 A

��with critical indices are

generators of the principal eigencone




an eigenvalue of A

the greatest (principal) eigenvalue of Athe only eigenvalue of A whose corresponding eigenvectorsmay be positivethe unique eigenvalue if A is irreducible (in this case alleigenvectors are positive)








an eigenvalue of Athe greatest (principal) eigenvalue of A

the only eigenvalue of A whose corresponding eigenvectorsmay be positivethe unique eigenvalue if A is irreducible (in this case alleigenvectors are positive)








an eigenvalue of Athe greatest (principal) eigenvalue of Athe only eigenvalue of A whose corresponding eigenvectorsmay be positive

the unique eigenvalue if A is irreducible (in this case alleigenvectors are positive)








an eigenvalue of Athe greatest (principal) eigenvalue of Athe only eigenvalue of A whose corresponding eigenvectorsmay be positivethe unique eigenvalue if A is irreducible (in this case alleigenvectors are positive)






Brief overview of tropical spectral theory: associated andcritical graph

A 2 Rn�n+ ,N = f1, ..., ng

A �! DA = (N,E ) ... associated graph

E = f(i , j); aij > 0g

A �! CA = (N,Ec ) ... critical graph

Ec ... the set of arcs of all critical cycles



A 2 Rn�n+ ,N = f1, ..., ngA �! DA = (N,E ) ... associated graph

E = f(i , j); aij > 0g

A �! CA = (N,Ec ) ... critical graph

Ec ... the set of arcs of all critical cycles



A 2 Rn�n+ ,N = f1, ..., ngA �! DA = (N,E ) ... associated graph

E = f(i , j); aij > 0g

A �! CA = (N,Ec ) ... critical graphEc ... the set of arcs of all critical cycles


Brief overview of tropical spectral theory: Example 1

A =

0BBBBBB@

0 2 0 5 3 08 0 2 0 0 00 0 0 0 0 10 0 5 0 2 00 2 0 0 4 83 0 0 4 0 0

1CCCCCCA , λ (A) = 4


Example 1: associated graph


Example 1: critical graph


Brief overview of tropical spectral theory: nding alleigenvalues

Frobenius Normal Form (FNF):

A t

0BBBBBB@A11A21 A22 0...

. . ....

. . .Ar1 Ar2 � � � � � � Arr

1CCCCCCA

A11, ...,Arr irreducibleThe corresponding partition of N : N1, ...,Nr ... classesReduced digraph Red(A) (partially ordered set):

nodes: 1, ..., rarcs: (i , j) : (9k 2 Ni )(9` 2 Nj )ak ` > 0

Ni �! Nj or i �! j means: there is a directed path from ito j in Red(A)




A t

0BBBBBB@A11A21 A22 0...

. . ....

. . .Ar1 Ar2 � � � � � � Arr

1CCCCCCAA11, ...,Arr irreducible

The corresponding partition of N : N1, ...,Nr ... classesReduced digraph Red(A) (partially ordered set):






A t

0BBBBBB@A11A21 A22 0...

. . ....

. . .Ar1 Ar2 � � � � � � Arr

1CCCCCCAA11, ...,Arr irreducibleThe corresponding partition of N : N1, ...,Nr ... classes

Reduced digraph Red(A) (partially ordered set):






A t

0BBBBBB@A11A21 A22 0...

. . ....

. . .Ar1 Ar2 � � � � � � Arr

1CCCCCCAA11, ...,Arr irreducibleThe corresponding partition of N : N1, ...,Nr ... classesReduced digraph Red(A) (partially ordered set):






A t

0BBBBBB@A11A21 A22 0...

. . ....

. . .Ar1 Ar2 � � � � � � Arr


nodes: 1, ..., r

arcs: (i , j) : (9k 2 Ni )(9` 2 Nj )ak ` > 0





A t

0BBBBBB@A11A21 A22 0...

. . ....

. . .Ar1 Ar2 � � � � � � Arr






A in an FNF:0BBBBBB@A11A21 A22 0...

. . ....

. . .Ar1 Ar2 � � � � � � Arr

1CCCCCCA , A11, ...,Arr irreducible

Spectral Theorem [linear] (Frobenius 1912, Victory 1985,Schneider, 1986):

Λ(A) = fλ(Aii );λ(Aii ) > λ(Ajj ) if j �! igSpectral Theorem [max] (Gaubert, Bapat, 1992):

Λ(A) = fλ(Aii );λ(Aii ) � λ(Ajj ) if j �! ig




. . ....

. . .Ar1 Ar2 � � � � � � Arr

1CCCCCCA , A11, ...,Arr irreducibleSpectral Theorem [linear] (Frobenius 1912, Victory 1985,Schneider, 1986):

Λ(A) = fλ(Aii );λ(Aii ) > λ(Ajj ) if j �! ig

Spectral Theorem [max] (Gaubert, Bapat, 1992):





. . ....

. . .Ar1 Ar2 � � � � � � Arr

1CCCCCCA , A11, ...,Arr irreducibleSpectral Theorem [linear] (Frobenius 1912, Victory 1985,Schneider, 1986):

Λ(A) = fλ(Aii );λ(Aii ) > λ(Ajj ) if j �! igSpectral Theorem [max] (Gaubert, Bapat, 1992):



Cyclicity of a matrix (index of imprimitivity)

Cyclicity of a strongly connected digraph = g.c.d. of thelengths of its cycles

Cyclicity of a digraph = l.c.m. of cyclicities of its SCCsCyclicity of A ... σ (A) or just σ

σ (A) = cyclicity of DA [linear algebra]σ (A) = cyclicity of CA [max algebra]




Cyclicity of a digraph = l.c.m. of cyclicities of its SCCs

Cyclicity of A ... σ (A) or just σ






σ (A) = cyclicity of DA [linear algebra]

σ (A) = cyclicity of CA [max algebra]


The Cyclicity Theorem

Cyclicity Theorem (Cohen et al, 1985):For every irreducible matrix A 2 Rn�n+ the cyclicity of A is theperiod of A, that is, the smallest natural number p for which thereis an integer T (A) such that

Ak+p = (λ (A))p Ak

for every k � T (A).T (A) ... transient of A


2.Matrix orbits


Setting for orbits:�

R[n

εdef= �∞

o,max,+

�

Given A 2 Rn�n and x 2 Rn the orbit of A (with startingvector x) is:

x ,A x ,A2 x ,A3 x ...Stability (steady regime):

A z = λ z , z 6= ε

Reachability of a stable regime:

Given A 2 Rn�n and x 2 Rn will the orbit reach a vectorz 6= ε satisfying A z = λ z?Attraction space: all x for which this is true



R[n

εdef= �∞

o,max,+

�Given A 2 Rn�n and x 2 Rn the orbit of A (with startingvector x) is:

x ,A x ,A2 x ,A3 x ...

Stability (steady regime):

A z = λ z , z 6= ε





R[n

εdef= �∞

o,max,+



A z = λ z , z 6= ε





R[n

εdef= �∞

o,max,+



A z = λ z , z 6= ε


Given A 2 Rn�n and x 2 Rn will the orbit reach a vectorz 6= ε satisfying A z = λ z?

Attraction space: all x for which this is true



R[n

εdef= �∞

o,max,+



A z = λ z , z 6= ε




Strongly and weakly stable matrices

V (A)... the set of all eigenvectors (Eig)

V (A) � attr (A) � Rn � fεg

Two extremes:

attr (A) = Rn � fεg ... A strongly stable (robust)

attr (A) = V (A) ... A weakly stable


Strongly and weakly stable matrices

V (A)... the set of all eigenvectors (Eig)

V (A) � attr (A) � Rn � fεgTwo extremes:

attr (A) = Rn � fεg ... A strongly stable (robust)

attr (A) = V (A) ... A weakly stable


Problem: Characterise strongly and weakly stable matrices


A irreducible:

A is strongly stable if and only if A is primitiveA is weakly stable if and only if the critical graph of A is aHamiltonian cycle in the associated graph

A reducible?


A irreducible:

A is strongly stable if and only if A is primitive

A is weakly stable if and only if the critical graph of A is aHamiltonian cycle in the associated graph

A reducible?


A irreducible:

A is strongly stable if and only if A is primitiveA is weakly stable if and only if the critical graph of A is aHamiltonian cycle in the associated graph

A reducible?


Strong stability (robustness)

Robustness criterion for reducible matrices (PB &S.Gaubert & RACG 2009):A with FNF classes N1, ...Nr and no ε column is robust if andonly if

All nontrivial classes are primitive and spectral(8i , j) If Ni ,Nj are non-trivial, Ni 9 Nj and Nj 9 Ni then

λ(Aii ) = λ(Ajj )




All nontrivial classes are primitive and spectral

(8i , j) If Ni ,Nj are non-trivial, Ni 9 Nj and Nj 9 Ni then

λ(Aii ) = λ(Ajj )




All nontrivial classes are primitive and spectral(8i , j) If Ni ,Nj are non-trivial, Ni 9 Nj and Nj 9 Ni then

λ(Aii ) = λ(Ajj )



Reduced digraph of a robust matrix with λ1 < λ2 < λ3 < λ4 :


Weakly stable matrices

A weakly stable: attr (A) = V (A)

Let A be irreducible

V (A) = fx 2 Rn;A x = λ (A) x , x 6= εg ...eigenvectorsV� (A) = fx 2 R

n;A x � λ (A) x , x 6= εg ...

subeigenvectorsV � (A) = fx 2 Rn;A x � λ (A) x , x 6= εg ...supereigenvectors





V (A) = fx 2 Rn;A x = λ (A) x , x 6= εg ...eigenvectors

V� (A) = fx 2 Rn;A x � λ (A) x , x 6= εg ...







n;A x � λ (A) x , x 6= εg ...

subeigenvectors

V � (A) = fx 2 Rn;A x � λ (A) x , x 6= εg ...supereigenvectors






n;A x � λ (A) x , x 6= εg ...




V (A) � V�(A) � Attr (A)

V (A) � V �(A) � Attr (A)A weakly stable =) V (A) = V �(A) = V�(A) = Attr (A)



V (A) � V�(A) � Attr (A)V (A) � V �(A) � Attr (A)

A weakly stable =) V (A) = V �(A) = V�(A) = Attr (A)



V (A) � V�(A) � Attr (A)V (A) � V �(A) � Attr (A)A weakly stable =) V (A) = V �(A) = V�(A) = Attr (A)



Theorem (PB+Sergeev, 2011)

Let A be irreducible.A is weakly stable () CA is a Hamilton cycle in DA.

0BBBB@��

�

1CCCCA



Theorem (PB+Sergeev, 2011)

A is weakly stable if and only if every spectral class of A is initialand weakly stable


3.The core (actually two cores) of a matrix


Our aim

Two semirings:

(R+,+, .)(R+,�, .) = (R+,max, .)

Aim: to provide a unied treatment of matrix powers inboth semirings based on a common combinatorialbackgroundHence deduce periodicity properties of matrix powers also inthe nonnegative linear case


Our aim

Two semirings:

(R+,+, .)

(R+,�, .) = (R+,max, .)



Our aim

Two semirings:

(R+,+, .)(R+,�, .) = (R+,max, .)



Our aim

Two semirings:

(R+,+, .)(R+,�, .) = (R+,max, .)

Aim: to provide a unied treatment of matrix powers inboth semirings based on a common combinatorialbackground

Hence deduce periodicity properties of matrix powers also inthe nonnegative linear case


Our aim

Two semirings:

(R+,+, .)(R+,�, .) = (R+,max, .)



Combinatorial background

The common combinatorial background:Theorem (Dulmage and Mendelsohn 1967, Brualdi andLewin 1982): For every integer k � 1 the set of nodes of astrongly connected digraph D with cyclicity σ is the union ofgcd (k, σ) pairwise disjoint sets of nodes of strongly connectedcomponents of Dk . (Digraph powers are Boolean)


Cones

[Statements below will cover both max and linear but only+ and . will be used - unless stated otherwise]

C � Rn+ is a cone if

x , y 2 C =) x + y 2 Cx 2 C , α � 0 =) αx 2 C


Cones


C � Rn+ is a cone ifx , y 2 C =) x + y 2 C

x 2 C , α � 0 =) αx 2 C


Cones


C � Rn+ is a cone ifx , y 2 C =) x + y 2 Cx 2 C , α � 0 =) αx 2 C


The core of a matrix

A 2 Rn�n+

In both max and lin algebra dene:

Im (A) = fAx ; x 2 Rn+g (cone)V (A,λ) = fx 2 Rn+;Ax = λxg , λ > 0 (eigencone)

Hence V (A,λ) � Im(A) for every λ > 0Im(A) � Im(A2) � Im(A3) � ...Core(A) def=

Tk�1

Im(Ak ) = ???

(Hence "two cores" dened in the same way)



A 2 Rn�n+In both max and lin algebra dene:



Tk�1

Im(Ak ) = ???





Im (A) = fAx ; x 2 Rn+g (cone)

V (A,λ) = fx 2 Rn+;Ax = λxg , λ > 0 (eigencone)


Tk�1

Im(Ak ) = ???







Tk�1

Im(Ak ) = ???






Hence V (A,λ) � Im(A) for every λ > 0

Im(A) � Im(A2) � Im(A3) � ...Core(A) def=

Tk�1

Im(Ak ) = ???






Hence V (A,λ) � Im(A) for every λ > 0Im(A) � Im(A2) � Im(A3) � ...

Core(A) def=Tk�1

Im(Ak ) = ???







Tk�1

Im(Ak ) = ???



Two cores of a nonnegative matrix

We have:

V (A,λ) � Im(A) for every λ > 0∑λ

V (A,λ)| {z }V Σ(A)

� Im(A)

V (A,λ) � V�Ak ,λk

�, k = 1, 2, ...

V Σ(A) � V Σ�Ak�� Im(Ak ), k = 1, 2, ...

V Σ(A) � Tk�1

Im(Ak ) = Core(A)

But also: V Σ(Ak ) � Core(A), k = 1, 2, ...



We have:

V (A,λ) � Im(A) for every λ > 0

∑λ

V (A,λ)| {z }V Σ(A)

� Im(A)


�, k = 1, 2, ...

V Σ(A) � V Σ�Ak�� Im(Ak ), k = 1, 2, ...

V Σ(A) � Tk�1

Im(Ak ) = Core(A)




We have:

V (A,λ) � Im(A) for every λ > 0∑λ

V (A,λ)| {z }V Σ(A)

� Im(A)


�, k = 1, 2, ...

V Σ(A) � V Σ�Ak�� Im(Ak ), k = 1, 2, ...

V Σ(A) � Tk�1

Im(Ak ) = Core(A)



Two cores of a nonnegative matrix - results

Theorem [both in max and linear]: Let A 2 Rn�n+ andσ = σ (A) . For every integer k � 1 :

V�Ak ,λk

�� V (Aσ,λσ)

V�Ak ,λk

�= V

�Ak+σ,λk+σ

�Theorem [both in max and linear]:

Core(A) = ∑k�1

V Σ�Ak�= V Σ (Aσ)

Linear: Pullman 1971Max: BSS 2012 (proof covers linear as well)Theorem: If gcd (k, σ) = gcd (l , σ) thenV Σ

�Ak�= V Σ

�Al�.




V�Ak ,λk

�� V (Aσ,λσ)

V�Ak ,λk

�= V

�Ak+σ,λk+σ

�

Theorem [both in max and linear]:

Core(A) = ∑k�1



�Ak�= V Σ

�Al�.




V�Ak ,λk

�� V (Aσ,λσ)

V�Ak ,λk

�= V

�Ak+σ,λk+σ


Core(A) = ∑k�1



�Ak�= V Σ

�Al�.




V�Ak ,λk

�� V (Aσ,λσ)

V�Ak ,λk

�= V

�Ak+σ,λk+σ


Core(A) = ∑k�1


Linear: Pullman 1971

Max: BSS 2012 (proof covers linear as well)Theorem: If gcd (k, σ) = gcd (l , σ) thenV Σ

�Ak�= V Σ

�Al�.




V�Ak ,λk

�� V (Aσ,λσ)

V�Ak ,λk

�= V

�Ak+σ,λk+σ


Core(A) = ∑k�1


Linear: Pullman 1971Max: BSS 2012 (proof covers linear as well)

Theorem: If gcd (k, σ) = gcd (l , σ) thenV Σ

�Ak�= V Σ

�Al�.




V�Ak ,λk

�� V (Aσ,λσ)

V�Ak ,λk

�= V

�Ak+σ,λk+σ


Core(A) = ∑k�1



�Ak�= V Σ

�Al�.


A property of the core(s)

Theorem [both in max and linear]: A induces a mapping onCore(A). This mapping permutes extremals of Core(A).

How to nd the core?Theorem [max only] The set of normalised extremals ofCore(A) is obtained by taking the distinct critical columns of��

λ�1A�σλ�� for all eigenvalues λ where σλ = lcm of

cyclicities of critical graphs for λ.



Theorem [both in max and linear]: A induces a mapping onCore(A). This mapping permutes extremals of Core(A).How to nd the core?

Theorem [max only] The set of normalised extremals ofCore(A) is obtained by taking the distinct critical columns of��





Theorem [both in max and linear]: A induces a mapping onCore(A). This mapping permutes extremals of Core(A).How to nd the core?Theorem [max only] The set of normalised extremals ofCore(A) is obtained by taking the distinct critical columns of��




Finite stabilisation

Finite stabilisation ... If there exists a k such that

Im(A) � Im(A2) � ... � Im(Ak ) = Im(Ak+1) = ...

Theorem [max only]: Finite stabilisation occurs for columnperiodic matrices, that is when for every i�

Ak+σi�.i= λσii A

k.i

for large k and some λi and σi .Column periodic matrices include orbit periodic matrices,that is when for every x 2 Rn+, x 6= 0 the orbit

�Akx

k=1,...

hits an eigenvector of some power of A.Orbit periodic matrices include ultimately periodicmatrices, that is when

Ak+σ = λσAk

for some σ and λ and all su¢ ciently large k.These include irreducible matrices.




Im(A) � Im(A2) � ... � Im(Ak ) = Im(Ak+1) = ...Theorem [max only]: Finite stabilisation occurs for columnperiodic matrices, that is when for every i�


k.i

for large k and some λi and σi .

Column periodic matrices include orbit periodic matrices,that is when for every x 2 Rn+, x 6= 0 the orbit

�Akx

k=1,...


Ak+σ = λσAk







k.i


�Akx

k=1,...

hits an eigenvector of some power of A.

Orbit periodic matrices include ultimately periodicmatrices, that is when

Ak+σ = λσAk







k.i


�Akx

k=1,...


Ak+σ = λσAk

for some σ and λ and all su¢ ciently large k.

These include irreducible matrices.






k.i


�Akx

k=1,...


Ak+σ = λσAk



THANK YOU!


Extremals of the cones

C � Rn+ ... a cone

u 2 C is an extremal if u = v +w (v �w in max), v ,w 2 Cimplies u = αv = βw for some α, β > 0

Any closed cone in Rn+ is generated by its extremals; inparticular, this holds for any nitely generated cone



C � Rn+ ... a coneu 2 C is an extremal if u = v +w (v �w in max), v ,w 2 Cimplies u = αv = βw for some α, β > 0

Any closed cone in Rn+ is generated by its extremals; inparticular, this holds for any nitely generated cone



The number of normalised extremals of an eigenconeV (A,λ) is

the number of strongly connected components of spectralclasses corresponding to λ in the associated graph (lin)the number of strongly connected components of the criticalgraph for λ (max)

σλ = lcm of cyclicities of spectral classes with λ (linear)σλ = lcm of cyclicities of critical graphs with λ (max)

The number of normalised extremals of V�Ak ,λk

�is

gcd (k, σλ)

σdef= lcm fσλ;λ 2 Λ(A)g

Every extremal of V�Ak ,λk

�is also in V

�Ark ,λrk

�for any

integer r � 1Theorem: If gcd (k, σ) = gcd (l , σ) then V Σ

�Ak�= V Σ

�Al�




the number of strongly connected components of spectralclasses corresponding to λ in the associated graph (lin)

the number of strongly connected components of the criticalgraph for λ (max)



�is

gcd (k, σλ)



�is also in V

�Ark ,λrk

�for any


�Ak�= V Σ

�Al�







�is

gcd (k, σλ)



�is also in V

�Ark ,λrk

�for any


�Ak�= V Σ

�Al�





σλ = lcm of cyclicities of spectral classes with λ (linear)

σλ = lcm of cyclicities of critical graphs with λ (max)


�is

gcd (k, σλ)



�is also in V

�Ark ,λrk

�for any


�Ak�= V Σ

�Al�







�is

gcd (k, σλ)



�is also in V

�Ark ,λrk

�for any


�Ak�= V Σ

�Al�







�is

gcd (k, σλ)



�is also in V

�Ark ,λrk

�for any

integer r � 1

Theorem: If gcd (k, σ) = gcd (l , σ) then V Σ�Ak�= V Σ

�Al�







�is

gcd (k, σλ)



�is also in V

�Ark ,λrk

�for any


�Ak�= V Σ

�Al�


If A is irreducible and primitive then by the Cyclicity Theorem:

Ak+1 = λ(A) Ak for k largeAk+1 x = λ(A) Ak x for k large and any x 2 Rn

A

�Ak x

�= λ(A)

�Ak x

�for k large and any

x 2 Rn

A irreducible: A is robust () A is primitive



Ak+1 = λ(A) Ak for k large

Ak+1 x = λ(A) Ak x for k large and any x 2 Rn

A

�Ak x

�= λ(A)

�Ak x


x 2 Rn




Ak+1 = λ(A) Ak for k largeAk+1 x = λ(A) Ak x for k large and any x 2 Rn

A

�Ak x

�= λ(A)

�Ak x


x 2 Rn



Documents

Tropical Linear Algebra: Orbits and the Core of a Matrixweb.mat.bham.ac.uk/P.Butkovic/files/Beamer Eindhoven.pdf · 2013. 3. 8. · Peter Butkoviµc (joint work with H. Schneider