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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Passages between max and lin
S.Friedland (1984):
A ... an irreducible nonnegative matrix
nAko∞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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Tropicalisation
Line segment
xy = fαx + βy ; α, β � 0, α+ β = 1g
Tropical line segment
xy = fαx � βy ; α, β � 0, α� β = 1g
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Tropicalisation
Line segment
xy = fαx + βy ; α, β � 0, α+ β = 1g
Tropical line segment
xy = fαx � βy ; α, β � 0, α� β = 1g
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extension to matrices and vectors
A� B = (aij � bij )
A B =�∑�k aik bkj
�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extension to matrices and vectors
A� B = (aij � bij )A B =
�∑�k aik bkj
�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Basic problems
One-sided max-linear systems (fast algorithms exist):
A x = b
A x � bA x = λxA x � λx
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Basic problems
One-sided max-linear systems (fast algorithms exist):
A x = bA x � b
A x = λxA x � λx
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Basic problems
One-sided max-linear systems (fast algorithms exist):
A x = bA x � bA x = λx
A x � λx
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Basic problems
One-sided max-linear systems (fast algorithms exist):
A x = bA x � bA x = λxA x � λx
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Basic problems
Two-sided max-linear systems (no polynomial method isknown):
A x = B xA x = B y
A x � c = B x � dA x = λB x (generalized eigenproblem)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Basic problems
Two-sided max-linear systems (no polynomial method isknown):
A x = B xA x = B yA x � c = B x � d
A x = λB x (generalized eigenproblem)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Basic problems
Two-sided max-linear systems (no polynomial method isknown):
A x = B xA x = B yA x � c = B x � dA x = λB x (generalized eigenproblem)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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,...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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,...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Basic problems
Periodicity of matrix powers:A,A2,A3, ...
Periodicity of matrix orbits:A x ,A2 x ,A3 x , ...Tropical characteristic polynomial and equation
Linear independence, regularity, rank,...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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,...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 π
The permanent is called strong if = for exactly one π
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Linear dependence/independence
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
∑�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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Linear dependence/independence
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
∑�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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Linear dependence/independence
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
∑�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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Linear dependence/independence
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
∑�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 =) LI
Strong regularity =) Gondran-Minoux regularity =)regularity
Strong rank � GM rank � Rank
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Linear dependence/independence
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
∑�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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Linear dependence/independence
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
∑�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 � 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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 cycle
Indices on critical cycles are critical indices
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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) � 1
λ�(λ (A))�1 A
�= 1
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: the principaleigenvalue
(Cuninghame-Green, 1962) For any A, λ(A) is
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)
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 A
the 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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 positive
the 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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: associated andcritical graph
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: associated andcritical graph
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: associated andcritical graph
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: associated andcritical graph
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Example 1: associated graph
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Example 1: critical graph
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
Frobenius Normal Form (FNF):
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):
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
Frobenius Normal Form (FNF):
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):
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
Frobenius Normal Form (FNF):
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):
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
Frobenius Normal Form (FNF):
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):
nodes: 1, ..., r
arcs: (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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
Frobenius Normal Form (FNF):
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):
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
Frobenius Normal Form (FNF):
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):
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
A in an FNF:0BBBBBB@A11A21 A22 0...
. . ....
. . .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):
Λ(A) = fλ(Aii );λ(Aii ) � λ(Ajj ) if j �! ig
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Brief overview of tropical spectral theory: nding alleigenvalues
A in an FNF:0BBBBBB@A11A21 A22 0...
. . ....
. . .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):
Λ(A) = fλ(Aii );λ(Aii ) � λ(Ajj ) if j �! ig
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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]
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 SCCs
Cyclicity of A ... σ (A) or just σ
σ (A) = cyclicity of DA [linear algebra]σ (A) = cyclicity of CA [max algebra]
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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]
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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]
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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]
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
2.Matrix orbits
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Problem: Characterise strongly and weakly stable matrices
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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?
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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?
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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?
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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?
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 )
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 )
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 )
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Strong stability (robustness)
Reduced digraph of a robust matrix with λ1 < λ2 < λ3 < λ4 :
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 ...eigenvectors
V� (A) = fx 2 Rn;A x � λ (A) x , x 6= εg ...
subeigenvectorsV � (A) = fx 2 Rn;A x � λ (A) x , x 6= εg ...supereigenvectors
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 ...
subeigenvectors
V � (A) = fx 2 Rn;A x � λ (A) x , x 6= εg ...supereigenvectors
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Weakly stable matrices
V (A) � V�(A) � Attr (A)
V (A) � V �(A) � Attr (A)A weakly stable =) V (A) = V �(A) = V�(A) = Attr (A)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Weakly stable matrices
V (A) � V�(A) � Attr (A)V (A) � V �(A) � Attr (A)
A weakly stable =) V (A) = V �(A) = V�(A) = Attr (A)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Weakly stable matrices
V (A) � V�(A) � Attr (A)V (A) � V �(A) � Attr (A)A weakly stable =) V (A) = V �(A) = V�(A) = Attr (A)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Weakly stable matrices
Theorem (PB+Sergeev, 2011)
Let A be irreducible.A is weakly stable () CA is a Hamilton cycle in DA.
0BBBB@����
�
1CCCCA
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Weakly stable matrices
Theorem (PB+Sergeev, 2011)
A is weakly stable if and only if every spectral class of A is initialand weakly stable
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
3.The core (actually two cores) of a matrix
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Cones
[Statements below will cover both max and linear but only+ and . will be used - unless stated otherwise]
C � Rn+ is a cone ifx , y 2 C =) x + y 2 C
x 2 C , α � 0 =) αx 2 C
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Cones
[Statements below will cover both max and linear but only+ and . will be used - unless stated otherwise]
C � Rn+ is a cone ifx , y 2 C =) x + y 2 Cx 2 C , α � 0 =) αx 2 C
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 > 0
Im(A) � Im(A2) � Im(A3) � ...Core(A) def=
Tk�1
Im(Ak ) = ???
(Hence "two cores" dened in the same way)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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)
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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, ...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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, ...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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, ...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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, ...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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, ...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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, ...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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, ...
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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�.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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�.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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�.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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�.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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 1971
Max: BSS 2012 (proof covers linear as well)Theorem: If gcd (k, σ) = gcd (l , σ) thenV Σ
�Ak�= V Σ
�Al�.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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�.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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�.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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.
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
THANK YOU!
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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 � 1
Theorem: If gcd (k, σ) = gcd (l , σ) then V Σ�Ak�= V Σ
�Al�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
Extremals of the cones
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�
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
If A is irreducible and primitive then by the Cyclicity Theorem:
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
�for k large and any
x 2 Rn
A irreducible: A is robust () A is primitive
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013