1. Three different ways There are three different ways to show that ρ(A) is a simple eigenvalue of an irreducible nonnegative matrix A:

10. Exercise Suppose that If A is irreducible and A ≠B, then

12. Remark such that with equality iff the nonzero w i s have the argument i.e.

13. Lemma 2.4.24 (Wielandt’s Lemma) p.1 Let Suppose that A is irreducible, nonnegative, andThen (a)

14. Lemma 2.4.24 (Wielandt’s Lemma) p.2 (b) Suppose that C has an eigenvalue then C is of the form where D is a diagonal matrix that satisfies

18. Exercise 2.4.25 p.1 and let k be a positive integer. (i) Prove that where the equality holds if and only if

20. Exercise 2.4.25 p.2 the algebraic multiplicity of (i) Prove also that when the equality holds, algebraic multiplicity of as an eigenvalue of A is equal to the as an eigenvalue of A.

22. Exercise 2.4.25 p.3 and Perron’s theorem to deduce that if A (ii) Use the result of part (i), Theorem 2.4.4 is a simple eigenvalue of A. is an irreducible nonnegative matrix, then

24. Exercise 2.4.26 p.1 Prove that if(i) Let and if A is irreducible Hint: Use Remark 2.4.9. then

26. Exercise 2.4.26 p.2 if A is an irreducible nonnegative matrix, (ii) Use the result of part (i) to deduce that different from A. for any principal submatrix B of A, then

28. Exercise 2.4.27 p.1 relation where y,z are Perron vectors of A and Letbe an irreducible nonnegative matrix. (i) Let B(t) =adj(tI-A). By using the multiplicity of ρ(A) as an eigenvalue of A is 1, at t=ρ(A) and the fact that the geometric show that B(ρ(A)) is of the form respectively, and c is a nonzero real constant.

30. Exercise 2.4.27 p.2 is the (n-1)-square principal the result of part (i) to deduce that (ii)Show that and column. (These relations are true for a where and hence submatrix obtained from A by deleting its row general square complex matrix A.) Then use is a simple eigenvalue of A.

34. Exercise 2.4.27 p.3 the largest real root of (iii) Using the fact that cannot be negative. Deduce that is equal to explain why

36. Exercise 2.4.28 matrix. Let B(t) have the same meaning as Let Exercise 2.4.26 (ii) and the relation be an irreducible nonnegative given in Exercise 2.4.27. Use the result of (which was established) to show that

38. Exercise 2.4.29 p.1 Let A be an irreducible nonnegative matrix with index of imprimitivity h (i) Prove that trace(A)=0 whenever h>1

42. Exercise 2.4.29 p.2 Let A be an irreducible nonnegative matrix with index of imprimitivity h (ii) Prove that h is a divisor of the number of nonzero eigenvalues of A (counting algebraic multiplicities)

44. Exercise 2.4.29 p.3 Let A be an irreducible nonnegative matrix with index of imprimitivity h (iii) Prove that if A is nonsingular, nxn and n is a prime, then h=1 or n.

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

47. Theorem2.4.22 (2 nd part of the Frobenius Thm) (ii) (iii) If Apply Wielandt’s Lemma to prove

48. Another proof in next page

51. Fully Cyclic The peripheral spectrum of A is called fully cyclic if then is an eigenvector of A corresponding to for all integers k.

52. Fully Cyclic p.2 Note that for k=0, the latter condition becomes

53. Corollary 2.4.30 Let A be an irreducible nonnegative matrix then the index of imprimitivity of A is equal to the spectral index of A.

55. Corollary 2.4.31 The peripheral spectrum of an irreducible nonnegative matrix is fully cyclic

57. Exercise 2.4.32 that the peripheral spectrum of P is fully If Let P be a square nonnegative matrix. Prove cyclic if and only if P satisfies the following: z is a corresponding eigenvector, then is a peripheral eigenvalue of P and

59. Theorem 2.4.33 If then there is a permutation matrix P such that whereis 1x1 or is irreducible. Frobenius normal form

61. Example

62. V 1 V 2 V 3 Three components all are maximal Strongly connected component

63. Remark p.1 If A is reducible, nonnegative, and there is a permutation matrix P such that whereis 1x1 or is irreducible.

64. Remark p.2 then form of The peripheral eigenvalue of A is of the times a root of unity and

65. Theorem 2.4.34 Given A is primitive if and only if

68. Exercise 2.4.35 The spectral radius of a nonnegative matrix A is positive if and only if G(A) contains at least a circuit.

69. Exercise 2.4.36 (i) Find the Frobenius normal form of the matrix (ii) Compute

70. Exercise 2.4.37 matrix and let vector x such that Let A be an nxn irreducible nonnegative and a positive Then there exists where is defined to be

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

72. A ⊕ B p.1

73. A ⊕ B p.2

74. Circuit Geometric mean circuit is called circuit geometric mean

75. Lemma 2.4.38 Let A be an irreducible nonnegative matrix x is a semipositive vector and such that then where μ(A) is the maximum circuit geometric mean. γ is called a max eigenvalue and x is the corresponding max eigenvector

79. Plus max algebra R ∪ {-∞} They are isometric

80. Brouwer’s Fixed Point Theorem Let C be a compact convex subset of and f is a continuous map from C to C, then

81. Theorem 2.4.39 Max version of the Perron-Frobenius Theorem If A be an irreducible nonnegative matrix then there is a positive vector x such that

83. Fact diagonal entries such that has the constant row sums. there is a diagonal matrix D with positive Given an irreducible nonnegative matrix A

84. Remark 2.4.40 diagonal entries such that in Theorem 2.4.39 is equivalent to the assertion there is a diagonal matrix D with positive that given an irreducible nonnegative matrix A the maximum entry in each row is the same.

86. If Let k be a fixed positive integer, the sum of the k largest, then let us define components in then

87. If Let k be a fixed positive integer, by, then let us define then

88. Theorem 2.4.41 and let constant Let A be an irreducible nonnegative matrix Then there exists a positive vector x and a such that

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

91. Combinatorial Spectral Theory of Nonnegative Matrices