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. D is cyclically 3-partite
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. Given counterexample in next page
Remark
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
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
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. It is stronger than “set”
Exercise 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 p.4
(d)´
m is a divisor of those indices i such that

51. 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.

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

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

55. 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.

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

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

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

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

63. 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 .

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

66. Proof of Theorem p.1

67. Proof of Theorem p.2

68. 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

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

75. 2.4 Irreducible Matrices