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

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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

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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

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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

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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

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MAX ALGEBRA a,b ≥ 0 a⊕b = max(a,b) a⊗b = ab

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 3 / 23

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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

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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

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same vs different a⊕b = 0 = ⇒ a = b = 0

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 4 / 23

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same vs different a⊕b = 0 = ⇒ a = b = 0 a⊕b = a = ⇒ b = 0

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 4 / 23

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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

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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

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max lin alg C = A⊕B : cij = aij ⊕bij C = A⊗B : cij =

  • k

aikbkj

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 5 / 23

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max lin alg C = A⊕B : cij = aij ⊕bij C = A⊗B : cij =

  • k

aikbkj 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

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max lin alg C = A⊕B : cij = aij ⊕bij C = A⊗B : cij =

  • k

aikbkj 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

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max lin alg C = A⊕B : cij = aij ⊕bij C = A⊗B : cij =

  • k

aikbkj 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

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where is the difference? A > 0,x y ≥ 0 = ⇒ Ax > Ay

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 6 / 23

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where is the difference? A > 0,x y ≥ 0 = ⇒ Ax > Ay A > 0,x y ≥ 0 = ⇒ A⊗x > A⊗y 2 2 1 2

  • ⊗

1 1

  • =

2 2

  • Hans Schneider

NONNEGATIVE MATRICES IN MAX ALGEBRA 6 / 23

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where is the difference? A > 0,x y ≥ 0 = ⇒ Ax > Ay A > 0,x y ≥ 0 = ⇒ A⊗x > A⊗y 2 2 1 2

  • ⊗

1 1

  • =

2 2

  • 2

2 1 2

  • ⊗

1

  • =

2 2

  • Hans Schneider

NONNEGATIVE MATRICES IN MAX ALGEBRA 6 / 23

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graph recap A ∈ Rn×n

+

, A ≥ 0

G(A): Graph of A

Vertex set {1,...,n} arcs i → j : aij > 0

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 7 / 23

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graph recap A ∈ Rn×n

+

, A ≥ 0

G(A): Graph of A

Vertex set {1,...,n} arcs i → j : aij > 0 i0

∗

→ ik : ∃(i1,...,ik−1) i0 → i1 ··· → ik−1 → ik

  • r

i = j

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 7 / 23

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graph recap A ∈ Rn×n

+

, A ≥ 0

G(A): Graph of A

Vertex set {1,...,n} arcs i → j : aij > 0 i0

∗

→ ik : ∃(i1,...,ik−1) i0 → i1 ··· → ik−1 → ik

  • r

i = j cycle γ: i0

∗

→ ik, i0 = ik

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 7 / 23

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graph recap A ∈ Rn×n

+

, A ≥ 0

G(A): Graph of A

Vertex set {1,...,n} arcs i → j : aij > 0 i0

∗

→ ik : ∃(i1,...,ik−1) i0 → i1 ··· → ik−1 → ik

  • r

i = j cycle γ: i0

∗

→ ik, i0 = ik cycle mean ¯ γ(A) = (ai0,i1 ···aik−1,ik)1/k ρ(A) = max¯ γ(A),γ(A) ∈ cG(A)

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 7 / 23

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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

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Critical graph Critical graph C(A): Graph induced in G(A) by the vertices of arcs lying on max cycles. A =           3/4 1 1/2 1/2 1 1 1 3/4 1/2 1 3/4 1/2 3/4 3/4          

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 8 / 23

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Critical graph Critical graph C(A): Graph induced in G(A) by the vertices of arcs lying on max cycles. A =           3/4 1 1/2 1/2 1 1 1 3/4 1/2 1 3/4 1/2 3/4 3/4           ρ(A) = 1 Components of C(A): {1,2,3,4},{5}

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 8 / 23

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Critical graph Critical graph C(A): Graph induced in G(A) by the vertices of arcs lying on max cycles. A =           3/4 1 1/2 1/2 1 1 1 3/4 1/2 1 3/4 1/2 3/4 3/4           ρ(A) = 1 Components of C(A): {1,2,3,4},{5}

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 8 / 23

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Very special?

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 9 / 23

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Very special? Not really!

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 9 / 23

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Very special? Not really! Observation: If B = X −1AX, where X is a pos diag matrix then bij = xiaij xj

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

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diagonal scaling Fiedler-Ptak(1967, 1969), M.Schneider -S (1990) Theorem Let A ∈ Rn×n

+

. There exists a pos diag X such that for B = X −1AX, bij = ρ(B) if (i,j) ∈ C(B) bij < ρ(B)

  • therwise

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 10 / 23

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diagonal scaling Fiedler-Ptak(1967, 1969), M.Schneider -S (1990) Theorem Let A ∈ Rn×n

+

. There exists a pos diag X such that for B = X −1AX, bij = ρ(B) if (i,j) ∈ C(B) bij < ρ(B)

  • therwise

We may assume matrix is strictly visualized

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 10 / 23

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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

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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

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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

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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

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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

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MX: Example A =           3/4 1 1/2 1/2 1 1 1 3/4 1/2 1 3/4 1/2 3/4 3/4           1 3/4 1 3/4 1 3/4 1 3/4 1/2 1 3/8 9/32 3/4 9/16 Two evectors of ρ(A) = 1

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 13 / 23

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Frobenius Normal Form collect strong conn cpts [classes] of G(A) and linearly order them After permutation similarity A =        A11 ... ... A21 A22 ... . . . . . . ... ... . . . . . . . . . ... Ak1 Ak2 ... ... Akk        each diagonal block irreducible

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 14 / 23

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reduced graph Reduced graph R (A) V = {1,,...,k} i → j ∈ E : Aij = 0 Path from i to j i0 → i1 → ··· → ip−1 → im Transitive closure R ∗(A) i

∗

→ j : exists path from i to j Skeleton S = R∗(A) (i,j) ∈ S : i

∗

→ k

∗

→ j implies k = i or k = j

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 15 / 23

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Example (1) ← − (2) ← − (3) ? տ ↑ (4)

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 16 / 23

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Example (1) ← − (2) ← − (3) ? տ ↑ (4)     ♠ ♥ ♠ ♦ ♥ ♠ ♦ ♥ ♠     ♠ irred block ♥ nonzero block ♦ in trans closure of skeleton

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 16 / 23

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Marked Reduced graph R (A) Vertex set {1,...,k} (classes) i → j ⇐ ⇒ Aij 0 j has access to i in R (A): i

∗

← j Each vertex marked with its (max) Perron root

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 17 / 23

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Distinguished, semi-distinguished vertices i distinguished i

∗

← j = ⇒ ρi > ρj

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 18 / 23

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Distinguished, semi-distinguished vertices i distinguished i

∗

← j = ⇒ ρi > ρj i semi-distinguished i

∗

← j = ⇒ ρi ≥ ρj

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 18 / 23

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Distinguished, semi-distinguished vertices i distinguished i

∗

← j = ⇒ ρi > ρj i semi-distinguished i

∗

← j = ⇒ ρi ≥ ρj (ρ1) ← − (ρ2) ← − (ρ3) ? տ ↑ (ρ4) ρ1 = ρ2 > ρ3 > ρ4

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 18 / 23

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Distinguished, semi-distinguished vertices i distinguished i

∗

← j = ⇒ ρi > ρj i semi-distinguished i

∗

← j = ⇒ ρi ≥ ρj (ρ1) ← − (ρ2) ← − (ρ3) ? տ ↑ (ρ4) ρ1 = ρ2 > ρ3 > ρ4 2,3,4 distinguished, 1 semi-distinguished

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 18 / 23

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MX: eigenvalues, eigenvectors Gaubert (1990s), Butkovic&Cuninghame-Green&Gaubert(2009) Theorem Let A be a nonnegative matrix in FNF. Then λ is an evalue

  • f (A) if snd only there is a semi-distinguished vertex i

with ρi = λ

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 19 / 23

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MX: eigenvalues, eigenvectors Gaubert (1990s), Butkovic&Cuninghame-Green&Gaubert(2009) Theorem Let A be a nonnegative matrix in FNF. Then λ is an evalue

  • f (A) if snd only there is a semi-distinguished vertex i

with ρi = λ The eigenvectors of A correspond to the semi-distinguished vertices of A: for for each semi-distinguished vertex i of R (A) there are (nonnegative) eigenvectors xi with Axi = ρixi such that xi

j > 0

if i ←← j xi

j = 0

  • therwise

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 19 / 23

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MX: eigenvalues, eigenvectors Gaubert (1990s), Butkovic&Cuninghame-Green&Gaubert(2009) Theorem Let A be a nonnegative matrix in FNF. Then λ is an evalue

  • f (A) if snd only there is a semi-distinguished vertex i

with ρi = λ The eigenvectors of A correspond to the semi-distinguished vertices of A: for for each semi-distinguished vertex i of R (A) there are (nonnegative) eigenvectors xi with Axi = ρixi such that xi

j > 0

if i ←← j xi

j = 0

  • therwise

Properly chosen, these form the exremals oif the cones of eignevectors

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 19 / 23

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NN: nonneg evals, evecs Theorem Let A be a nonnegative matrix in FNF. Then λ is an evalue

  • f (A) if and only if there is a -distinguished vertex i with

ρi = λ

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 20 / 23

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NN: nonneg evals, evecs Theorem Let A be a nonnegative matrix in FNF. Then λ is an evalue

  • f (A) if and only if there is a -distinguished vertex i with

ρi = λ The nonnegative eigenvectors of A correspond to the distinguish vertices of A: for for each distinguished vertex i of R (A) there is nonnegative eigenvector xi with Axi = ρixi such that xi

j > 0

if i ←← j xi

j = 0

  • therwise

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 20 / 23

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NN: nonneg evals, evecs Theorem Let A be a nonnegative matrix in FNF. Then λ is an evalue

  • f (A) if and only if there is a -distinguished vertex i with

ρi = λ The nonnegative eigenvectors of A correspond to the distinguish vertices of A: for for each distinguished vertex i of R (A) there is nonnegative eigenvector xi with Axi = ρixi such that xi

j > 0

if i ←← j xi

j = 0

  • therwise

Properly chosen, these are linearly independent, and for any part evalue, form the extremals of the cone of nonneg evectors

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 20 / 23

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NN example A =     4 1 4 1 3 2 3 2     [4] ← [5]∗∗ ← [2]∗∗ . 1 . 1 . 1 . 1

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 21 / 23

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MX example A =     4 1 4 1 3 2 3 2    

  • [4]∗

← [4]∗∗ ← [2]∗∗ 1 1/4 1 3/16 3/4 9/64 3/8 1

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 22 / 23

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P . Butkovic Max-Linear Systems: Theory and Algorithms Springer 2010

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 23 / 23

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P . Butkovic Max-Linear Systems: Theory and Algorithms Springer 2010 That’s it for today next time: Commuting matrices in three incarnations

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 23 / 23

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P . Butkovic Max-Linear Systems: Theory and Algorithms Springer 2010 That’s it for today next time: Commuting matrices in three incarnations THANKS!

Hans Schneider NONNEGATIVE MATRICES IN MAX ALGEBRA 23 / 23