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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
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1.Tropical linear algebra
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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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
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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
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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
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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
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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
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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
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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
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Example 1: associated graph
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Example 1: critical graph
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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2.Matrix orbits
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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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
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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
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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
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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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
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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
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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
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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
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Problem: Characterise strongly and weakly stable matrices
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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THANK YOU!
Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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Peter Butkoviµc (joint work with H. Schneider and S. Sergeev) Eindhoven 13 March 2013
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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
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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
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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
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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