1. Chapter 2 Nonnegative Matrices

2. 2-1 Introduction

3. Entrywise nonnegative (entrywise ) nonnengative means different from positive semidefinite

4. Strictly positive strictly positive means different from positive definite

5. Remark e.g. nonzero, nonnegative but not positive semipositive≡nonzero, nonnegative

8. Remark e.g.

9. 2-2 Perron’s Theorem

10. spectral radius spectral radius 譜半徑

11. Example

12. Proven in next page

14. Collatz Wielandt collatz weilandt

15. Lemma 2.2.2 (1) Proven in next page

17. Lemma 2.2.2 (2) Proven in next page ( 證明很重要 ) is closed and bounded above

20. Remark Proven in next page

22. generalized eigenvector u is called generalized eigenvector ofA if

23. Remark Proven in next page the columns of P are the generalized eigenvectors of A.

26. Remark The geometric multiple of λ =1 and there is no generalized eigenvector other than eigenvector corr. to λ

27. Remark Proven in next page

30. Remark Proven in next page

32. Remark If A>0, then A has no nonnegative eigenvector other than (multiple of) u, where u>0 and Proven in next page( 證明很特別 )

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

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

36. Norm on a vector space (i) (iii) (ii) is a norm on V = hold iff x=0

37. we introduce a metric is a metric space with on V, by

38. Convergent matrix sequence can be interpreted in where one of the following equivalent way: (i)

39. is in any fixed norm of where The topology of (ii) is independent of

40. (the maximum norm) to be we obtain (i) In (ii), take

41. Bounded matrix sequence (ii) (i) is bounded means

42. Fact 2.2.4 (ii) (i)

43. (iii)

44. Apply of Fact 2.2.4 (ii) and P is nonsigular If then convergent problem of A is corresponding to convergent problem of

45. Theorem 2.2.3 Let (i) The sequence converges to the zero matrix iff

46. (ii) converges iff or and 1 is the only eigenvalue with modulus 1 and the corresp. Jordan blocks are all

47. (iii) is bounded iff either and ifthen or

48. Lemma 2.2.5 (i) If (ii) If then and m=1, then

49. (iii) If the sequence and m=1, then is bounded Note: In this case, the seqence does not converge if explain in next page

50. θ

51. (iv) If then the sequence or is unbounded and

55. Exercise 2.2.7 eigenvalue and is non-nipotent for every Suppose that is a simple

56. what can you tell about the vector Prove that x and y? exists and is of the form

57. 2-3 Nonnegative Matrices

58. Lemma 2.2.2 is closed, bounded above and If, then

59. Lemma 2.3.1 If, then

62. Lemma 2.3.2 If, then

64. Fact

65. Corollary 2.3.3, and B is a principal submatrix of A If then In particular

67. Exercise 2.3.4 then If Hint: There is some α>1 such that

69. Theorem 2.3.5 (Perron-Frobenius Thm), thenIf and

72. R i (A) = i th row sum of A

73. C j (A) = j th column sum of A

74. Corollary 2.3.6 Then Let and

77. Matrix norm is called a matrix norm if N( - ) is a A norm N( - ) on norm on, and N( - ) is submultiplicative i.e.

78. Matrix norm Induced by Vector norm be a (vector) norm on Let Define onby matrix norm induced by the vector norm

79. Proposition of matrix norm induced by vector norm

81. Remark 2.3.7 is a marix norm on If then

82. not Euclidean matrix norm correct proof in next page ( 很重要 )

84. Special norm:l ∞, l p

85. Special Matrix norm be the matrix norm on Let induced by the norm of

86. Corollary If the row sums of A are constant Let then A row sum of A

88. Exercise 2.3.8 p.1 max absolute column sum of A

89. Exercise 2.3.8 p.2 max absolute row sum of A

92. Exercise 2.3.9 Prove that if A has a positive eigenvector, then the corresponding eigenvalue is Let [Hint: Apply the Perron-Frobenius Thm to A T ]

94. Remark 2.3.10 If A has equal row sums, then Let If A has equal column sums, then

97. a row stochastic matrix with row sums all equal to 1,then A is called a row stochastic matrix. If

98. a column stochastic matrix with column sums all equal to 1,then A is called a column stochastic matrix. If

99. Exercise 2.3.11 [ Hint: Let Deduce Corollary 2.3.6 from Remark 2.3.9 and Lemma 2.3.2 To show that inequality consider B=DA, where D is the diagonal matrix show that

104. Diagonally Similar p.1 are diagonal similar In particular if there is nonsingular matrix D s.t.

105. Diagonally Similar p.2 preserves the class of nonnegative (as well as, positive) matrices. In particular nonnegative diagonal similarity

106. Corollary 2.3.12 Then for any positive vector and we have

108. Exercise 2.3.13 p.1 For any semipositive vector Wielandt numbers of A with respect to x are defined and denoted respectively by: the upper and the lower Collatz- Let

109. Exercise 2.3.13 p.2 (we adopt the convention that inf ψ=∞) Prove that for any semipositive x, we have

110. Exercise 2.3.13

113. Exercise 2.3.14 p.1 (i) Prove that if Let for some positive vector x then

115. Exercise 2.3.14 p.2 (ii) Prove that if Let for some positive vector x then

117. Exercise 2.3.14 p.3 (iii) Use parts (i) and (ii) to deduce that Let if A has a positive eigenvector then the corresponding eigenvalue is

119. Exercise 2.3.15 p.1 are diagonally similar. Show that the matrices and

121. Exercise 2.3.15 p.2 are diagonally similar ? Are the matrices and