SPECTRAL THEORY OF REDUCIBLE NONNEGATIVE MATRICES IN MAX ALGEBRA - PowerPoint PPT Presentation

SPECTRAL THEORY OF REDUCIBLE NONNEGATIVE MATRICES IN MAX ALGEBRA Hans Schneider Chemnitz October 2010 maxspectchmn 21 Sept 2010, 14:30 Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 1 / 23

max, min, +, times max plus ( R , max , +) max times ( R + 0 , max , × ) min plus ( R , min , +) = tropical min times ( R + 0 , min , × ) Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 2 / 23

max, min, +, times max plus ( R , max , +) max times ( R + 0 , max , × ) min plus ( R , min , +) = tropical min times ( R + 0 , min , × ) We do max times Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 2 / 23

MAX ALGEBRA a , b ≥ 0 a ⊕ b = max ( a , b ) a ⊗ b = ab Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 3 / 23

MAX ALGEBRA a , b ≥ 0 a ⊕ b = max ( a , b ) a ⊗ b = ab ( R + 0 , ⊕ , ⊗ ) is a semiring: a commutative semigroup with 0 under max, and ⊗ distributes over ⊕ Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 3 / 23

MAX ALGEBRA a , b ≥ 0 a ⊕ b = max ( a , b ) a ⊗ b = ab ( R + 0 , ⊕ , ⊗ ) is a semiring: a commutative semigroup with 0 under max, and ⊗ distributes over ⊕ Just like ( R + 0 , + , × ) ? Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 3 / 23

same vs different a ⊕ b = 0 = ⇒ a = b = 0 Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 4 / 23

same vs different a ⊕ b = 0 = ⇒ a = b = 0 a ⊕ b = a � = ⇒ b = 0 Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 4 / 23

same vs different a ⊕ b = 0 = ⇒ a = b = 0 a ⊕ b = a � = ⇒ b = 0 a ⊕ b = a = ⇒ a ≥ b Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 4 / 23

same vs different a ⊕ b = 0 = ⇒ a = b = 0 a ⊕ b = a � = ⇒ b = 0 a ⊕ b = a = ⇒ a ≥ b a ⊕ a = a Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 4 / 23

max lin alg C = A ⊕ B : c ij = a ij ⊕ b ij � C = A ⊗ B : c ij = a ik b kj k Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 5 / 23

max lin alg C = A ⊕ B : c ij = a ij ⊕ b ij � C = A ⊗ B : c ij = a ik b kj k Stephane Gaubert 1997: The spectral theory in MX "is extremely similar to the well-known Perron-Frobenius theory" in NN Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 5 / 23

max lin alg C = A ⊕ B : c ij = a ij ⊕ b ij � C = A ⊗ B : c ij = a ik b kj k Stephane Gaubert 1997: The spectral theory in MX "is extremely similar to the well-known Perron-Frobenius theory" in NN with some important differences. Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 5 / 23

max lin alg C = A ⊕ B : c ij = a ij ⊕ b ij � C = A ⊗ B : c ij = a ik b kj k Stephane Gaubert 1997: The spectral theory in MX "is extremely similar to the well-known Perron-Frobenius theory" in NN with some important differences. Our aim is to compare and contrast the two theories Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 5 / 23

where is the difference? A > 0 , x � y ≥ 0 = ⇒ Ax > Ay Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 6 / 23

where is the difference? A > 0 , x � y ≥ 0 = ⇒ Ax > Ay A > 0 , x � y ≥ 0 � = ⇒ A ⊗ x > A ⊗ y � 2 � � 1 � � 2 � 2 ⊗ = 1 2 1 2 Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 6 / 23

where is the difference? A > 0 , x � y ≥ 0 = ⇒ Ax > Ay A > 0 , x � y ≥ 0 � = ⇒ A ⊗ x > A ⊗ y � 2 � � 1 � � 2 � 2 ⊗ = 1 2 1 2 � 2 � � 0 � � 2 � 2 ⊗ = 1 2 1 2 Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 6 / 23

graph recap A ∈ R n × n , A ≥ 0 + G ( A ) : Graph of A Vertex set { 1 ,..., n } arcs i → j : a ij > 0 Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 7 / 23

graph recap A ∈ R n × n , A ≥ 0 + G ( A ) : Graph of A Vertex set { 1 ,..., n } arcs i → j : a ij > 0 ∗ i 0 → i k : ∃ ( i 1 ,..., i k − 1 ) i 0 → i 1 ··· → i k − 1 → i k i = j or Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 7 / 23

graph recap A ∈ R n × n , A ≥ 0 + G ( A ) : Graph of A Vertex set { 1 ,..., n } arcs i → j : a ij > 0 ∗ i 0 → i k : ∃ ( i 1 ,..., i k − 1 ) i 0 → i 1 ··· → i k − 1 → i k i = j or ∗ → i k , i 0 = i k cycle γ : i 0 Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 7 / 23

graph recap A ∈ R n × n , A ≥ 0 + G ( A ) : Graph of A Vertex set { 1 ,..., n } arcs i → j : a ij > 0 ∗ i 0 → i k : ∃ ( i 1 ,..., i k − 1 ) i 0 → i 1 ··· → i k − 1 → i k i = j or ∗ → i k , i 0 = i k cycle γ : i 0 γ ( A ) = ( a i 0 , i 1 ··· a i k − 1 , i k ) 1 / k cycle mean ¯ ρ ( A ) = max ¯ γ ( A ) , γ ( A ) ∈ cG ( A ) Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 7 / 23

Critical graph Critical graph C ( A ) : Graph induced in G ( A ) by the vertices of arcs lying on max cycles. Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 8 / 23

Critical graph Critical graph C ( A ) : Graph induced in G ( A ) by the vertices of arcs lying on max cycles.   3 / 4 1 1 / 2 0 0 1 / 2 0 0 0 1 0 0 0 0     0 0 0 1 0 0 0     A = 1 0 0 0 3 / 4 0 0     1 / 2 0 0 0 1 3 / 4 0     1 / 2 0 0 0 0 0 0   3 / 4 0 0 0 0 3 / 4 0 Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 8 / 23

Critical graph Critical graph C ( A ) : Graph induced in G ( A ) by the vertices of arcs lying on max cycles.   3 / 4 1 1 / 2 0 0 1 / 2 0 0 0 1 0 0 0 0     0 0 0 1 0 0 0     A = 1 0 0 0 3 / 4 0 0     1 / 2 0 0 0 1 3 / 4 0     1 / 2 0 0 0 0 0 0   3 / 4 0 0 0 0 3 / 4 0 ρ ( A ) = 1 Components of C ( A ) : { 1 , 2 , 3 , 4 } , { 5 } Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 8 / 23

Critical graph Critical graph C ( A ) : Graph induced in G ( A ) by the vertices of arcs lying on max cycles.   3 / 4 1 1 / 2 0 0 1 / 2 0 0 0 1 0 0 0 0     0 0 0 1 0 0 0     A = 1 0 0 0 3 / 4 0 0     1 / 2 0 0 0 1 3 / 4 0     1 / 2 0 0 0 0 0 0   3 / 4 0 0 0 0 3 / 4 0 ρ ( A ) = 1 Components of C ( A ) : { 1 , 2 , 3 , 4 } , { 5 } Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 8 / 23

Very special? Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 9 / 23

Very special? Not really! Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 9 / 23

Very special? Not really! Observation: If B = X − 1 AX , where X is a pos diag matrix then b ij = x i a ij x j G ( B ) = cG ( A ) ¯ γ ( A ) = ¯ γ ( B ) , ∀ cycles γ ρ ( B ) = ρ ( A ) C ( B ) = cC ( A ) Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 9 / 23

diagonal scaling Fiedler-Ptak(1967, 1969), M.Schneider -S (1990) Theorem Let A ∈ R n × n . There exists a pos diag X such that for + B = X − 1 AX, b ij = ρ ( B ) if ( i , j ) ∈ C ( B ) b ij < ρ ( B ) otherwise Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 10 / 23

diagonal scaling Fiedler-Ptak(1967, 1969), M.Schneider -S (1990) Theorem Let A ∈ R n × n . There exists a pos diag X such that for + B = X − 1 AX, b ij = ρ ( B ) if ( i , j ) ∈ C ( B ) b ij < ρ ( B ) otherwise We may assume matrix is strictly visualized Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 10 / 23

NN: Perron-Frobenius for irred matrices Frobenius (1912) Theorem Let A ≥ 0 be irreducible. Then its spec rad ρ ( A ) is its the unique eigenvalue with an assoc nonneg eigenvector Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 11 / 23

NN: Perron-Frobenius for irred matrices Frobenius (1912) Theorem Let A ≥ 0 be irreducible. Then its spec rad ρ ( A ) is its the unique eigenvalue with an assoc nonneg eigenvector which is (ess) unique and positive. Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 11 / 23

NN: Perron-Frobenius for irred matrices Frobenius (1912) Theorem Let A ≥ 0 be irreducible. Then its spec rad ρ ( A ) is its the unique eigenvalue with an assoc nonneg eigenvector which is (ess) unique and positive. ρ ( A ) is the Perron root of A Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 11 / 23

MX: Perron-Frobenius for irred matrices Cunninghame-Greene (1960s) Theorem Let A ≥ 0 be irreducible. Then its max cyc mean ρ ( A ) is its unique (dist) eigenvalue. There is an (ess) unique associated positive eigenvector for each component of the crit graph, Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 12 / 23

MX: Perron-Frobenius for irred matrices Cunninghame-Greene (1960s) Theorem Let A ≥ 0 be irreducible. Then its max cyc mean ρ ( A ) is its unique (dist) eigenvalue. There is an (ess) unique associated positive eigenvector for each component of the crit graph, which are the extremals of the max cone of eigenvectors. ρ ( A ) will be called the (max) Perron root of A Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 12 / 23