Combinatorial Spectral Theory of Nonnegative Matrices

Theorem 2.2.1 p.1 (Perron’s Thm)1907 (a) (b) (c)

(d) (e) A has no nonnegative eigenvector other than (multiples of) u. (f) (g)

Theorem 2.3.5 (Perron-Frobenius Thm) If , then and

Frobenius Thm (1912) Part I (Corollary 2.4.7) Let If A is irreducible, then the conclusions (a),(b),(c),(d) and (f) of Perron Thm all hold.

Frobenius Thm Part II p.1 Concerning the peripheral spectrum of P (表面譜) i.e.

Frobenius Thm Part II p.2 The usual proof of Part II of Frobenius Thm relies on Wielandt’s Lemma. Provide a different approach

Index of Imprimitivity D: strongly connected digraph : vertex set k=k(D): = g.c.d. of the length of the closed directed walks of D. k is called the index of inprimitivity of D.

Circuit and Cycle Circuit is a simple closed directed walk. Cycle is a simple closed walk . (usually used in graph not diagraph)

Note k(D) = g.c.d. of the circuit lengths of D Any strongly connected digraph has a circuit except for a single vertex.

Primitive or Imprimitive A digraph D is called primitive if k(D)=1 , and imprimitive if K(D)>1

Theorem 2.4.13 p.1 Let D be a strongly connected digraph of order n and k=k(D). Then can not write circuits (i) For any vertex k=g.c.d of lengths of closed directed walks containing a.

Theorem 2.4.13 p.2 (ii) For each pair of vertices a and b, the lengths of the directed walks from a to b are congruent modulo k. (iii) We can write such that

V2 V3 Vk V1 D is cyclically k-partite Vk+1 ≡V1 V1,V2 ,…,Vk are called the sets of imprimitivity of D

Theorem 2.4.13 p.4 (iv) For the length of a directed walk from is congruent to j-i mod k.

Exercise 2.4.14 Let D be a strongly connected digraph of order n and k=k(D). Then Show that for any vertex k=g.c.d of differences of lengths of directed walk from a to b .

V2 V3 V6 V1 D is cyclically 6-partite D is cyclically 2-partite and cyclically 3-partite

V1∪ V3∪ V5 V2∪ V4∪ V6 D is cyclically 2-partite

D is cyclically 3-partite V1∪ V4 V2∪ V5 V3∪ V6 D is cyclically 3-partite

Remark If D is cyclically r-partite, then D is cyclically s-partite if

Cyclic Index of a digraph Cyclic index of a digraph : = the largest integer r s.t. the digraph is cyclically r-partite

Theorem 2.4.15 Let D be a strongly connected digraph. Then cyclic index of D = index of imprimitivity of D Furthermore, D is cyclically r-partite iff r is a divisor of k(D).

Given counterexample in next page Remark 2.4.16 If D is a diagraph which is not strongly connected and if k is the g.c.d of circuit lengths of D, then D need not be cyclically k-partite. Given counterexample in next page

1 2 3 4 k=2 D is not cyclically r-partite for any r≧2

r-cyclic matrix in the superdiagonal block form A square matrix A is r-cyclic if G(A) is cyclically r-partite or equivalently permutation similar

G(A) 1 6 3 5 2 4 see next page

Cyclic Index of a Matrix Let A be a square matrix. Define cyclic index of A= cyclic index of G(A)

Remark 2.4.17 If A is r-cyclic, then A is diagonally similar to

Spectral Index If Denote is called the spectral index of A.

The Sum of KxK Principal Minors

Theorem 1.2.3

It is stronger than “set” Exercise 2.4.18 p.1 It is stronger than “set” see above page Let be a positive integer. Prove that the following conditions are equivalent: (a) (b) A and have the same char. poly.

Exercise 2.4.18 p.2 (c) The characteristic polynomial of A is of the form for some nonnegative p and some monic polynomial f with nonzero constant term.

Exercise 2.4.18 p.3 (d) Let where are different from zero and Then m divides the differences (or, equivalently, the differences

Exercise 2.4.18 p.4 (d)´ m is a divisor of those indices i such that

Remark The spectral index of A is equal to the largest positive integer m such that the equivalent conditions in the exercise are all satisfied.

Circuit Product Let and γ is a circuit in G(A) with arcs is called the circuit product of A w.r.t. γ

Remark 2.4.19 If are diagonally similar, then G(A)=G(B) and for each circuitγ in G(A), we have

Exercise 2.4.20 p.1 Let such that G(A)=G(B) and suppose that A (hence also B) is irreducible. Prove that if for each circuit γ in G(A). Then A and B are diagonally similar.

Exercise 2.4.20 p.2 Given an example to illustrate the result no longer hold if the irreducibility assumption is omitted.

Lemma 2.4.20 (Schur) Let be closed under addition and d:= g.c.d of elements of S, then such that

Frobenius-Schur Index Denote ψ(S) to be the smallest N such that Lemma 2.4.20 holds. ψ(S) is called Frobenius-Schur index of S

Frobenius-Schur Index If S is the set of all nonnegative linear combinations of the positive integers then denote

Lemma 2.4.21 Let D be a strongly connected digraph , k=index of imprimitivity of D, and are the sets of imprimitivity of D. Then such that for any there is a directed walk from xi to xj of length j-i+tk .

Theorem 1.2.13 If m=n ,and at least one of A or B is nonsigular,then AB and BA are similar

Proof of Theorem 1.2.13 p.1

Proof of Theorem 1.2.13 p.2

Theorem2.4.22 (2nd part of the Frobenius Thm) Given irreducible matrix with m distinct eigenvalues with moduloρ(A) Then (i) the peripheral spectrum of A is

Theorem2.4.22 (2nd part of the Frobenius Thm) (ii) (iii) If

2.4 Irreducible Matrices