2.5 Matrix With Cyclic Structure

1. 2.5 Matrix With Cyclic Structure

2. Remark When A is an irreducible nonnegative matrix # of distinct peripheral eigenvalues of A = the index of imprimitivity of G(A) = cyclic index of A = spectral index of A = the largest m such that

3. sgn(τ) A cyclic permutation transpositions. a product of

4. sgn(π) Any permutation can be writen of the form: where each is cyclic permuation.

7. Remark If G(A) has no circuits, then detA=0 , then there are (vertex-) If disjoint circuits in G(A), the sum of lengths is n.

8. Ek(A) = the sum of nonzero terms of the form (or ) are disjoint circuits where the sum of lengths is equal to k.

9. Remark If G(A) has no circuits, then then i.e. A is nilpotent

10. Example 2.5.1 G(A) 1 2 3 4

12. Example 2.5.2 G(A) 1 2 3 4

14. Cofactor of A Let If

16. Example by digraph Consider 6 4 3 1 2 5

18. Csr In general, is the sum of possible terms of the form or where αis a path from vertex r to vervex s and are circuits s.t. the path and the circuits are mutually disjoint and together they contain all vertices of G(A)

19. Dn complete digraph of order n Consider as an edge-weighted digraph circuit weight of

20. weight of D(π) permutation digraph product of weights of circuits

21. Example Let 1 4 5 2 3 then

22. Remark 2.5.5 (i) Let If A and B have the same set of circuits for each circuit and then A and B have equal corresponding principal minors of all possible orders and as a consequence

24. Fact cf. Exercise 2.4.20 why does A must be irreducible? see next second page.

25. 1 1 1 2 1 1 1 does not appear in circuit product 1 G(A)=G(B) is not irreducibole But A and B are not diagonally similar.

26. Remark 2.5.5 (ii) A and B have equal corresponding principal A and B have the same minors A Counter example circuit products. A and B have the same corresponding principal minors

27. Counter Example 1 1 1 1 1 2 2 G(A): G(B): 1 1 1 1 3 3

28. Counter Example 2 A and AT have the same principal minors But we may have

29. Question A and B have the same principal minors Hartfiel and Loewy proved the following:

31. Introduce A Semiring On introduce by: R+ form a semiring under 0 is zero element are associative,commutative distributes over

32. max-product algebra A⊕B p.1

33. A⊕B p.2

34. Fuzzy Matrix Version max-min algebra

35. Boolean Matrix B: (0,1)-matrix different from F2 Spectial case of max-min algebra

36. In max-product algebra, max-min algebra satisfies the associative low.

37. in the sence max algebra or fuzzy algebra

38. a directed walk of length two in G(A) from i to j

39. a directed walk of length k in G(A) from i to j Furthermore, G(A) contain the directed walk the sum of walk products of A w.r.t the directed walks in G(A) from i to j of length k

40. In the setting of Max algebra (max-product algebra) p.1 a directed walk of length two in G(A) from i to j the maximum of walk products of A w.r.t the directed walks in G(A) from i to j of length two

41. In the setting of Max algebra (max-product algebra) p.2 a directed walk of length k in G(A) from i to j the maximum of walk products of A w.r.t the directed walks in G(A) from i to j of length k

42. In the setting of Fuzzy Matrix (max-min algebra) a directed walk of length k in G(A) from i to j It is difficult to explain the geometric meaning of

43. In the setting of Fuzzy Matrix (max-min algebra) p.2 a directed walk of length k in G(A) from i to j It is difficult to explain the geometric meaning of Furthermore, from i to j of length k G(A) contain the directed walk

44. Combinatorial Spectral Theory of Nonnegative Matrices

45. Plus max alge They are isometric

46. Remark p.2 Let X be a Topology space, F is a Banach space and is continuous map such that T(X) is precompact in F. Then for any there is a continuous map of finite rank s.t.

47. sgn(π) A permutation where each is cyclic permuation.