SPECTRAL THEORY OF REDUCIBLE NONNEGATIVE MATRICES: A GRAPH - - PowerPoint PPT Presentation

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spectral theory of reducible nonnegative matrices a graph

SPECTRAL THEORY OF REDUCIBLE NONNEGATIVE MATRICES: A GRAPH - - PowerPoint PPT Presentation

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SPECTRAL THEORY OF REDUCIBLE NONNEGATIVE MATRICES: A GRAPH THEORETIC APPROACH Hans Schneider Chemnitz October 2010 reducible100920 version 20 Sep 2010 19:00 printed September 21, 2010 Hans Schneider Reducible nonnegative matrices 1 / 28

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SPECTRAL THEORY OF REDUCIBLE NONNEGATIVE MATRICES: A GRAPH THEORETIC APPROACH

Hans Schneider Chemnitz October 2010

reducible100920 version 20 Sep 2010 19:00 printed September 21, 2010 Hans Schneider Reducible nonnegative matrices 1 / 28

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Aim of talk After reviewing the classical Perron-Frobenius theory of irreducible matrices we turn to the reducible case and discuss it in terms of underlying graphs.

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Graph of A A ∈ Rnn

+ , A ≥ 0

G(A): Graph of A

Vertex set {1,...,n}

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Graph of A A ∈ Rnn

+ , A ≥ 0

G(A): Graph of A

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

∗

→ j : ∃(i1,...,ik) i → i1 → ··· → ik → j

r

i = j

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Irreducibility A irreducible:

G(A) strongly connected (∀i,j,

i

∗

→ j): ⇐ ⇒ NOT, after permutation similarity, A11 A12 A22

with A11,A22 square, really there

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Irreducibility A irreducible:

G(A) strongly connected (∀i,j,

i

∗

→ j): ⇐ ⇒ NOT, after permutation similarity, A11 A12 A22

with A11,A22 square, really there

(0) irreducible

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Irreducible Perron-Frobenius ρ(A) = max{|λ| : λ ∈ spec(A)} spectral radius of A ∈ Rnn

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Irreducible Perron-Frobenius ρ(A) = max{|λ| : λ ∈ spec(A)} spectral radius of A ∈ Rnn Perron (1907, 1907) Frobenius (1908, 1909, 1912) Theorem A ≥ 0, irreducible, THEN 0 < ρ(A) ∈ spec(A), (A = (0))

Hans Schneider Reducible nonnegative matrices 5 / 28

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Irreducible Perron-Frobenius ρ(A) = max{|λ| : λ ∈ spec(A)} spectral radius of A ∈ Rnn Perron (1907, 1907) Frobenius (1908, 1909, 1912) Theorem A ≥ 0, irreducible, THEN 0 < ρ(A) ∈ spec(A), (A = (0)) ρ(A) simple eigenvalue

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Irreducible Perron-Frobenius ρ(A) = max{|λ| : λ ∈ spec(A)} spectral radius of A ∈ Rnn Perron (1907, 1907) Frobenius (1908, 1909, 1912) Theorem A ≥ 0, irreducible, THEN 0 < ρ(A) ∈ spec(A), (A = (0)) ρ(A) simple eigenvalue ∃ unique x, Ax = ρx, & x > 0

Hans Schneider Reducible nonnegative matrices 5 / 28

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Irreducible Perron-Frobenius ρ(A) = max{|λ| : λ ∈ spec(A)} spectral radius of A ∈ Rnn Perron (1907, 1907) Frobenius (1908, 1909, 1912) Theorem A ≥ 0, irreducible, THEN 0 < ρ(A) ∈ spec(A), (A = (0)) ρ(A) simple eigenvalue ∃ unique x, Ax = ρx, & x > 0 x is the only nonnegative evector

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By continuity Theorem A ≥ 0 THEN ρ(A) ∈ spec(A),

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By continuity Theorem A ≥ 0 THEN ρ(A) ∈ spec(A), ∃x 0, Ax = ρx

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By continuity Theorem A ≥ 0 THEN ρ(A) ∈ spec(A), ∃x 0, Ax = ρx   1  

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By continuity Theorem A ≥ 0 THEN ρ(A) ∈ spec(A), ∃x 0, Ax = ρx   1   Much, much more may be said about reducible nonneg A

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Frobenius Normal Form (FNF) collect strong conn cpts of G(A)

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

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

R (A): Reduced Graph of A

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

∗

→ j in R (A)

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

R (A): Reduced Graph of A

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

∗

→ j in R (A) partial order of classes

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Marked reduced graph Each vertex marked with its Perron root (spec rad) Example     A11 · · · A22 · · A31 A32 A33 · ? ? A43 A44    

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Marked reduced graph Each vertex marked with its Perron root (spec rad) Example     A11 · · · A22 · · A31 A32 A33 · ? ? A43 A44     (ρ1) (ρ2) \ / (ρ3) | (ρ4) ρi = ρ(Aii)

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QUESTIONS Nonnegativity of eigenvectors Nonnegativity of generalized eigenvectors: (A−λI)kx = 0 Nonnegativity of basis for generalized eigenspace for ρ(A) Nonnegativity of Jordan basis for ρ Relation of Jordan form to graph structure for ρ

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QUESTIONS Nonnegativity of eigenvectors Nonnegativity of generalized eigenvectors: (A−λI)kx = 0 Nonnegativity of basis for generalized eigenspace for ρ(A) Nonnegativity of Jordan basis for ρ Relation of Jordan form to graph structure for ρ We explore how the nonnegativity, combinatorial, spectral properties inter-relate, see e.g. LAA 84 (1986), 161 - 189.

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Frobenius 1912, Victory 1985 Definition Vertex i of is a R (A) is a distinguished vertex if i

∗

← j = ⇒ ρi > ρj Theorem Let A be a nonnegative matrix in FNF. Then the nonnegative eigenvectors of A correspond to the distinguish vertices of A:

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Frobenius 1912, Victory 1985 Definition Vertex i of is a R (A) is a distinguished vertex if i

∗

← j = ⇒ ρi > ρj Theorem Let A be a nonnegative matrix in FNF. Then the nonnegative eigenvectors of A correspond to the distinguish vertices of A: 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

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Frobenius 1912, Victory 1985 Definition Vertex i of is a R (A) is a distinguished vertex if i

∗

← j = ⇒ ρi > ρj Theorem Let A be a nonnegative matrix in FNF. Then the nonnegative eigenvectors of A correspond to the distinguish vertices of A: 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

These are linearly independent, and for any part evalue, extremals of the cone of nonneg evectors. (Carlson 1963)

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Example     A11 · · · A22 · · A31 A32 A33 · ? ? A43 A44     ρ1 > ρ3 = ρ4 > ρ2 (ρ1)∗∗ (ρ2) \ / (ρ3)∗ | (ρ4)∗∗

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continuation ρ1 > ρ3 = ρ4 > ρ2 (ρ1)∗∗ (ρ2) \ / (ρ3)∗ | (ρ4)∗∗ ρ1 ρ4 + + + +

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Warning! Nonnegative eigenvectors!   · · · 1 1   (0) (0) \ / (0) Eigenvectors   1     1 −1  

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Reminder: Jordan Form Jordan block (of size 4):     λ 1 λ 1 λ 1 λ     Theorem Over the complex numbers, every matrix is similar to a direct sum of Jordan blocks. indλ(A) : = max size of J–block for λ = min{k : N = N (λI −A)k+1 = N (λI −A)k}

N – generalized nullspace of A

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Reminder: Jordan Form Jordan block (of size 4):     λ 1 λ 1 λ 1 λ     Theorem Over the complex numbers, every matrix is similar to a direct sum of Jordan blocks. indλ(A) : = max size of J–block for λ = min{k : N = N (λI −A)k+1 = N (λI −A)k}

N – generalized nullspace of A

Q: Does the red graph determine the J-form for ρ?

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Special case 1: S 1952/56 Theorem A ∈ Rnn

+ . TFAE:

(a) dim(N (A−ρI)) = 1 (a’) All Jordan block for ρ are size 1 (b) The ρ classes are trivially ordered. (a) & (a’) are complex algebra (b) is combinatorial

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Example     A11 · · · A22 · · A31 A32 A33 · ? ? A43 A44     (ρ1)∗∗ (ρ2)∗∗ \ / (ρ3) | (ρ4) (ρ =) ρ1 = ρ2 > ρ3 = ρ4 J-form for ρ is (1,1)

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continuation (ρ1)∗∗ (ρ2)∗∗ \ / (ρ3) | (ρ4) + + + + + + These are the only evecs for ρ

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Special case 2: S 1952/56 Theorem A ∈ Rnn

+ . TFAE:

(a) dim null(A−ρI) = multρ(A) (a’) There is only one Jordan block for ρ (b) The ρ classes are linearly ordered.

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Example (ρ1)∗∗ (ρ2) \ / (ρ3) | (ρ4)∗∗ (ρ =) ρ1 = ρ4 > ρ2 = ρ3 x z + + + + (ρI −A)x = z, (ρI −A)z = 0

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Example (ρ1)∗∗ (ρ2) \ / (ρ3) | (ρ4)∗∗ (ρ =) ρ1 = ρ4 > ρ2 = ρ3 x z + + + + (ρI −A)x = z, (ρI −A)z = 0 J-form form for ρ is (2)

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General case?? Example that stopped me in 1952     · · · . · · 1 1 · a 1 ·    

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General case?? Example that stopped me in 1952     · · · . · · 1 1 · a 1 ·     × Jordan form a = 1 (2,2) a = 1 (2,1,1)

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General case?? Example that stopped me in 1952     · · · . · · 1 1 · a 1 ·     × Jordan form a = 1 (2,2) a = 1 (2,1,1) Hershkowitz-S (1991) "Solved" the problem using majorization

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Rothblum index thm 1975 indρ(A) : = max size of J–block for A = min{k : N = N (ρI −A)k+1 = N (ρI −A)k} Theorem indρ = max length of chain of ρ classes

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Example     · · · . · · 1 1 · a 1 ·    

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Example     · · · . · · 1 1 · a 1 ·     × max chain of 0 classes = 2

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Example     · · · . · · 1 1 · a 1 ·     × max chain of 0 classes = 2 Jordan form: either (2,2) or (2,1,1) either case ind0 = 2

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generalized eigenvectors x a gen evector of A for λ (A−λI)rx = 0, r > 0

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generalized eigenvectors x a gen evector of A for λ (A−λI)rx = 0, r > 0

Nλ(A) := {x : (A−λI)rx = 0, r ≥ n}

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generalized eigenvectors x a gen evector of A for λ (A−λI)rx = 0, r > 0

Nλ(A) := {x : (A−λI)rx = 0, r ≥ n}

i is a semi-distinguished vertex: i

∗

← j = ⇒ ρi ≥ ρj

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Preferred basis Theorem Rothblum(1975), Richman-S(1978), Hershkowitz-S(1988) Theorem Let λ ≥ 0. Suppose the semi-dist vertices of A with ρi = λ are i1 < ... < is. Then there exist xp, p = 1,...,s in N (A) such that xp

j > 0

if ip

∗

← j xp

j = 0

therwise

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Preferred basis Theorem Rothblum(1975), Richman-S(1978), Hershkowitz-S(1988) Theorem Let λ ≥ 0. Suppose the semi-dist vertices of A with ρi = λ are i1 < ... < is. Then there exist xp, p = 1,...,s in N (A) such that xp

j > 0

if ip

∗

← j xp

j = 0

therwise

and such that (A−λI)xp = ∑

q

cpqxq where cpq > 0 if ip

∗

← iq, q = p cpq = 0

therwise

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Example     · · · . · · 1 1 · 1 1 ·    

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Example     · · · . · · 1 1 · 1 1 ·     0∗ 0∗ × 0∗∗ 0∗∗

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Example     · · · . · · 1 1 · 1 1 ·     0∗ 0∗ × 0∗∗ 0∗∗ x1 x2 x3 x4 1 1 1 1 1 1 1 1

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Example     · · · . · · 1 1 · 1 1 ·     0∗ 0∗ × 0∗∗ 0∗∗ x1 x2 x3 x4 1 1 1 1 1 1 1 1 Ax1 = Ax2 = x3 +x4 Ax3 = Ax4 = 0

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more     · · · . · · 1 1 · 1 1 ·     Ax1 = Ax2 = x3 +x4 Ax3 = Ax4 = 0

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more     · · · . · · 1 1 · 1 1 ·     Ax1 = Ax2 = x3 +x4 Ax3 = Ax4 = 0 These vectors span N0 but are not lin indep Rothblum(1975) Theorem The gen null space for ρ(A) has a nonneg basis

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proofs? By Frobenius tracedown method: Solve successively equations for xi ≥ 0 of the form (Aii −ρiIii)xi = bi where bi ≥ 0.

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proofs? By Frobenius tracedown method: Solve successively equations for xi ≥ 0 of the form (Aii −ρiIii)xi = bi where bi ≥ 0. Carlson1963 Ax +b = ρx given reducible A ≥ 0 and b ≥ 0.

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two references H.S The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and related properties: A survey, Lin. Alg. Appl. 84 (1986), 161-189.

D. Hershkowitz and H.S, On the existence of matrices

with prescribed height and level characteristics, Israel Math J. 75 (1991), 105-117.

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two references H.S The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and related properties: A survey, Lin. Alg. Appl. 84 (1986), 161-189.

D. Hershkowitz and H.S, On the existence of matrices

with prescribed height and level characteristics, Israel Math J. 75 (1991), 105-117. THANK YOU

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