Geometric Perron-Frobenius theory in finite dimensions Hans - PowerPoint PPT Presentation

Geometric Perron-Frobenius theory in finite dimensions Hans Schneider Chemnitz October 2010 geompfchmn 2010.09.08, 11:05 September 8, 2010 Hans Schneider Geometric Perrron-Frobenius 1 / 14

cones Definition K a cone in R n K + K ⊆ K 1 ( x , y ∈ K = ⇒ x + y ∈ K ) R + K ⊆ K 2 ( α ≥ 0 , x ∈ K = ⇒ α x ∈ K K pointed 3 K ∩ K = { 0 } ( x , − x ∈ K = ⇒ x = 0 ) Hans Schneider Geometric Perrron-Frobenius 2 / 14

cones Definition K a cone in R n K + K ⊆ K 1 ( x , y ∈ K = ⇒ x + y ∈ K ) R + K ⊆ K 2 ( α ≥ 0 , x ∈ K = ⇒ α x ∈ K K pointed 3 K ∩ K = { 0 } ( x , − x ∈ K = ⇒ x = 0 ) We assume K closed (in Euclidean topology of R n ) Hans Schneider Geometric Perrron-Frobenius 2 / 14

proper cones Definition cone K proper: span ( K ) = K − K = R n ( ∀ z ∈ R n ∃ x , y ∈ K , z = x − y ) Hans Schneider Geometric Perrron-Frobenius 3 / 14

proper cones Definition cone K proper: span ( K ) = K − K = R n ( ∀ z ∈ R n ∃ x , y ∈ K , z = x − y ) proper cone p.o’s R n y ≥ x : y − x ∈ K Hans Schneider Geometric Perrron-Frobenius 3 / 14

proper cones Definition cone K proper: span ( K ) = K − K = R n ( ∀ z ∈ R n ∃ x , y ∈ K , z = x − y ) proper cone p.o’s R n y ≥ x : y − x ∈ K x ≥ 0 ⇐ ⇒ x ∈ K x > 0 ⇐ ⇒ x ∈ int K Hans Schneider Geometric Perrron-Frobenius 3 / 14

faces Definition F a face of cone K −− F � K : F is a subcone of K and x ∈ F , 0 ≤ y ≤ x = ⇒ y ∈ F Hans Schneider Geometric Perrron-Frobenius 4 / 14

faces Definition F a face of cone K −− F � K : F is a subcone of K and x ∈ F , 0 ≤ y ≤ x = ⇒ y ∈ F F a proper face of K : F � = 0 , F � = K Hans Schneider Geometric Perrron-Frobenius 4 / 14

three examples R n + Faces: (+ , 0 , + 0 ,..., 0 , +) Hans Schneider Geometric Perrron-Frobenius 5 / 14

three examples R n + Faces: (+ , 0 , + 0 ,..., 0 , +) ( x 2 1 + x 2 2 + ··· + x 2 n − 1 ) ( 1 / 2 ) ≤ x n Proper faces: ( x 2 1 + x 2 2 + ··· x 2 n − 1 ) ≤ x n Extremals Hans Schneider Geometric Perrron-Frobenius 5 / 14

three examples R n + Faces: (+ , 0 , + 0 ,..., 0 , +) ( x 2 1 + x 2 2 + ··· + x 2 n − 1 ) ( 1 / 2 ) ≤ x n Proper faces: ( x 2 1 + x 2 2 + ··· x 2 n − 1 ) ≤ x n Extremals (Real) Space: Hermitian n × n matrices Cone: Positive semi-definite matrices Faces: Matrices simult similar to 0 k ⊕ X Hans Schneider Geometric Perrron-Frobenius 5 / 14

nonnegative operators K a proper cone in R n A ≥ 0 : AK ⊆ K A > 0 : AK ⊆ int ( K ) Hans Schneider Geometric Perrron-Frobenius 6 / 14

nonnegative operators K a proper cone in R n A ≥ 0 : AK ⊆ K A > 0 : AK ⊆ int ( K ) F an invariant face: AF ⊆ F Hans Schneider Geometric Perrron-Frobenius 6 / 14

nonnegative operators K a proper cone in R n A ≥ 0 : AK ⊆ K A > 0 : AK ⊆ int ( K ) F an invariant face: AF ⊆ F A ≥ 0 irreducible: A leaves no proper face of K invariant: ⇒ ( I + A ) n − 1 > 0 ⇐ Hans Schneider Geometric Perrron-Frobenius 6 / 14

Aim of talk The aim is to relate the structure of invariant faces of the cone to the spectral properties of an operator nonnegative w.r.t. the cone. Hans Schneider Geometric Perrron-Frobenius 7 / 14

Aim of talk The aim is to relate the structure of invariant faces of the cone to the spectral properties of an operator nonnegative w.r.t. the cone. As we related the graph theoretic and spectral properties of a nonnegative matrix both in classical nonneg alg and max alg. Hans Schneider Geometric Perrron-Frobenius 7 / 14

fin dim Krein-Rutman 1940’s Theorem A ≥ 0 irreducible spec rad ρ ( A ) ∈ spec ( A ) ρ ( A ) simple eigenvalue ∃ ! x > 0 , Ax = ρ x y � 0 , Ay = λ y = ⇒ λ = ρ , y = x Hans Schneider Geometric Perrron-Frobenius 8 / 14

fin dim Krein-Rutman 1940’s Theorem A ≥ 0 irreducible spec rad ρ ( A ) ∈ spec ( A ) ρ ( A ) simple eigenvalue ∃ ! x > 0 , Ax = ρ x y � 0 , Ay = λ y = ⇒ λ = ρ , y = x And for reducible matrices? Hans Schneider Geometric Perrron-Frobenius 8 / 14

RECALL combinatorial theory ρ i := ρ ( A ii ) ∗ i is a distinguished vertex : i ← j = ⇒ ρ i > ρ j Theorem Let A be a nonnegative matrix in FNF. Then the nonnegative eigenvectors of A correspond to the distinguish vertices of A: for for each distinguished vertex i of ∆( A ) there is nonnegative eigenvector x i with Ax i = ρ i x i such that ∗ x i j > 0 i ← j if x i j = 0 otherwise These are linearly independent, and all others are nonneg lin combs. Hans Schneider Geometric Perrron-Frobenius 9 / 14

distinguished faces F � K : F an invariant face of K ρ F = ρ ( A | F ) F � K distinguished: F , G invariant, G ⊳ F = ⇒ ρ G < ρ F Theorem K a proper cone in R n . Each distinguished face contains an eigenvector of ρ F in its relative interior. For any particular eigenvalue, the eigenvectors so obtained are the extremals of the cone of nonnegative eigenvectors. Hans Schneider Geometric Perrron-Frobenius 10 / 14

chains of semi-dist inv faces F � K is semi-distinguished : G � F , G invariant = ⇒ ρ G ≤ ρ F Hans Schneider Geometric Perrron-Frobenius 11 / 14

chains of semi-dist inv faces F � K is semi-distinguished : G � F , G invariant = ⇒ ρ G ≤ ρ F A ∈ R n × n in FNF + ind λ ( A ) : = max size of J–block for λ = min { k : N = N ( λ I − A ) k + 1 = N ( λ I − A ) k } Hans Schneider Geometric Perrron-Frobenius 11 / 14

chains of semi-dist inv faces F � K is semi-distinguished : G � F , G invariant = ⇒ ρ G ≤ ρ F A ∈ R n × n in FNF + ind λ ( A ) : = max size of J–block for λ = min { k : N = N ( λ I − A ) k + 1 = N ( λ I − A ) k } Invariant F λ -face: ρ F = λ K � F 1 ⊲ F 2 ⊲ ··· ⊲ F : proper chain of inv semi-dist λ faces ord λ = max length of such a chain of faces Hans Schneider Geometric Perrron-Frobenius 11 / 14

generalization of Rothblum’s theorem Tam-S (2001) Theorem ind λ ≤ ord λ If K is polyhedral (fin gen), ind ρ = ord ρ Hans Schneider Geometric Perrron-Frobenius 12 / 14

generalization of Rothblum’s theorem Tam-S (2001) Theorem ind λ ≤ ord λ If K is polyhedral (fin gen), ind ρ = ord ρ If K = R n + this is a generalization by Hershkowitz-S of Rothblum’s theorem (case λ = ρ ) Hans Schneider Geometric Perrron-Frobenius 12 / 14

Tam-S, Matrices leaving a cone invariant , Art. 26, Handbook of Linear Algebra (ed. L. Hogben) 2007 Hans Schneider Geometric Perrron-Frobenius 13 / 14

Tam-S, Matrices leaving a cone invariant , Art. 26, Handbook of Linear Algebra (ed. L. Hogben) 2007 THANK YOU And please allow me one more slide! Hans Schneider Geometric Perrron-Frobenius 13 / 14

Hans Schneider Geometric Perrron-Frobenius 14 / 14