Combinatorial Spectral Theory of Nonnegative Matrices

1. Combinatorial Spectral Theory of Nonnegative Matrices

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

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

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

5. 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.

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

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

8. 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.

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

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

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

13. 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.

15. 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

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

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

21. 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 .

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

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

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

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

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

28. 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).

30. 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

31. 1 2 3 4 k=2 D is not cyclically r-partite for any r≧2

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

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

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

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

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

40. The Sum of KxK Principal Minors

41. Theorem 1.2.3

43. 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.

44. 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.

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

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