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
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. Hans Schneider Reducible nonnegative matrices 2 / 28
Graph of A A ∈ R nn + , A ≥ 0 G ( A ) : Graph of A Vertex set { 1 ,..., n } Hans Schneider Reducible nonnegative matrices 3 / 28
Graph of A A ∈ R nn + , A ≥ 0 G ( A ) : Graph of A Vertex set { 1 ,..., n } i → j : a ij > 0 ∗ i → j : ∃ ( i 1 ,..., i k ) i → i 1 → ··· → i k → j or i = j Hans Schneider Reducible nonnegative matrices 3 / 28
Irreducibility A irreducible: ∗ G ( A ) strongly connected ( ∀ i , j , i → j ): ⇐ ⇒ NOT, after permutation similarity, � A 11 � 0 A 12 A 22 with A 11 , A 22 square, really there Hans Schneider Reducible nonnegative matrices 4 / 28
Irreducibility A irreducible: ∗ G ( A ) strongly connected ( ∀ i , j , i → j ): ⇐ ⇒ NOT, after permutation similarity, � A 11 � 0 A 12 A 22 with A 11 , A 22 square, really there (0) irreducible Hans Schneider Reducible nonnegative matrices 4 / 28
Irreducible Perron-Frobenius ρ ( A ) = max {| λ | : λ ∈ s pec ( A ) } spectral radius of A ∈ R nn Hans Schneider Reducible nonnegative matrices 5 / 28
Irreducible Perron-Frobenius ρ ( A ) = max {| λ | : λ ∈ s pec ( A ) } spectral radius of A ∈ R nn 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
Irreducible Perron-Frobenius ρ ( A ) = max {| λ | : λ ∈ s pec ( A ) } spectral radius of A ∈ R nn Perron (1907, 1907) Frobenius (1908, 1909, 1912) Theorem A ≥ 0 , irreducible, THEN 0 < ρ ( A ) ∈ spec ( A ) , ( A � = ( 0 )) ρ ( A ) simple eigenvalue Hans Schneider Reducible nonnegative matrices 5 / 28
Irreducible Perron-Frobenius ρ ( A ) = max {| λ | : λ ∈ s pec ( A ) } spectral radius of A ∈ R nn 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 = ρ x , & x > 0 Hans Schneider Reducible nonnegative matrices 5 / 28
Irreducible Perron-Frobenius ρ ( A ) = max {| λ | : λ ∈ s pec ( A ) } spectral radius of A ∈ R nn 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 = ρ x , & x > 0 x is the only nonnegative evector Hans Schneider Reducible nonnegative matrices 5 / 28
By continuity Theorem A ≥ 0 THEN ρ ( A ) ∈ spec ( A ) , Hans Schneider Reducible nonnegative matrices 6 / 28
By continuity Theorem A ≥ 0 THEN ρ ( A ) ∈ spec ( A ) , ∃ x � 0 , Ax = ρ x Hans Schneider Reducible nonnegative matrices 6 / 28
By continuity Theorem A ≥ 0 THEN ρ ( A ) ∈ spec ( A ) , ∃ x � 0 , Ax = ρ x 0 0 0 1 0 0 0 0 0 Hans Schneider Reducible nonnegative matrices 6 / 28
By continuity Theorem A ≥ 0 THEN ρ ( A ) ∈ spec ( A ) , ∃ x � 0 , Ax = ρ x 0 0 0 1 0 0 0 0 0 Much, much more may be said about reducible nonneg A Hans Schneider Reducible nonnegative matrices 6 / 28
Frobenius Normal Form (FNF) collect strong conn cpts of G ( A ) Hans Schneider Reducible nonnegative matrices 7 / 28
Frobenius Normal Form (FNF) collect strong conn cpts of G ( A ) After permutation similarity A 11 0 ... ... 0 A 21 A 22 0 ... 0 . . . ... ... . . . A = . . . . . ... . . . . 0 A k 1 A k 2 ... ... A kk each diagonal block irreducible Hans Schneider Reducible nonnegative matrices 7 / 28
Frobenius Normal Form (FNF) collect strong conn cpts of G ( A ) After permutation similarity A 11 0 ... ... 0 A 21 A 22 0 ... 0 . . . ... ... . . . A = . . . . . ... . . . . 0 A k 1 A k 2 ... ... A kk each diagonal block irreducible R ( A ) : Reduced Graph of A Vertex set { 1 ,..., k } (classes) i → j ⇐ ⇒ A ij � 0 ∗ i has access to j in R ( A ) : i → j in R ( A ) Hans Schneider Reducible nonnegative matrices 7 / 28
Frobenius Normal Form (FNF) collect strong conn cpts of G ( A ) After permutation similarity A 11 0 ... ... 0 A 21 A 22 0 ... 0 . . . ... ... . . . A = . . . . . ... . . . . 0 A k 1 A k 2 ... ... A kk each diagonal block irreducible R ( A ) : Reduced Graph of A Vertex set { 1 ,..., k } (classes) i → j ⇐ ⇒ A ij � 0 ∗ i has access to j in R ( A ) : i → j in R ( A ) partial order of classes Hans Schneider Reducible nonnegative matrices 7 / 28
Marked reduced graph Each vertex marked with its Perron root (spec rad) Example A 11 · · · 0 A 22 · · A 31 A 32 A 33 · ? ? A 43 A 44 Hans Schneider Reducible nonnegative matrices 8 / 28
Marked reduced graph Each vertex marked with its Perron root (spec rad) Example A 11 · · · 0 A 22 · · A 31 A 32 A 33 · ? ? A 43 A 44 ( ρ 1 ) ( ρ 2 ) \ / ( ρ 3 ) | ( ρ 4 ) ρ i = ρ ( A ii ) Hans Schneider Reducible nonnegative matrices 8 / 28
QUESTIONS Nonnegativity of eigenvectors Nonnegativity of generalized eigenvectors: ( A − λ I ) k x = 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 Hans Schneider Reducible nonnegative matrices 9 / 28
QUESTIONS Nonnegativity of eigenvectors Nonnegativity of generalized eigenvectors: ( A − λ I ) k x = 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. Hans Schneider Reducible nonnegative matrices 9 / 28
Frobenius 1912, Victory 1985 Definition Vertex i of is a R ( A ) is a distinguished vertex if ∗ 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: Hans Schneider Reducible nonnegative matrices 10 / 28
Frobenius 1912, Victory 1985 Definition Vertex i of is a R ( A ) is a distinguished vertex if ∗ 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 each distinguished vertex i of R ( 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 Hans Schneider Reducible nonnegative matrices 10 / 28
Frobenius 1912, Victory 1985 Definition Vertex i of is a R ( A ) is a distinguished vertex if ∗ 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 each distinguished vertex i of R ( 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 for any part evalue, extremals of the cone of nonneg evectors. (Carlson 1963) Hans Schneider Reducible nonnegative matrices 10 / 28
Example A 11 · · · 0 A 22 · · A 31 A 32 A 33 · ? ? A 43 A 44 ρ 1 > ρ 3 = ρ 4 > ρ 2 ( ρ 1 ) ∗∗ ( ρ 2 ) \ / ( ρ 3 ) ∗ | ( ρ 4 ) ∗∗ Hans Schneider Reducible nonnegative matrices 11 / 28
continuation ρ 1 > ρ 3 = ρ 4 > ρ 2 ( ρ 1 ) ∗∗ ( ρ 2 ) \ / ( ρ 3 ) ∗ | ( ρ 4 ) ∗∗ ρ 1 ρ 4 + 0 0 0 + 0 + + Hans Schneider Reducible nonnegative matrices 12 / 28
Warning! Nonnegative eigenvectors! · · 0 0 0 · 1 1 0 ( 0 ) ( 0 ) \ / ( 0 ) Eigenvectors 0 1 0 − 1 1 0 Hans Schneider Reducible nonnegative matrices 13 / 28
Reminder: Jordan Form Jordan block (of size 4): λ 1 0 0 0 λ 1 0 0 0 λ 1 0 0 0 λ Theorem Over the complex numbers, every matrix is similar to a direct sum of Jordan blocks. ind λ ( A ) : = max size of J–block for λ = min { k : N = N ( λ I − A ) k + 1 = N ( λ I − A ) k } N – generalized nullspace of A Hans Schneider Reducible nonnegative matrices 14 / 28
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