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Chapter 2 Nonnegative Matrices
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2-1 Introduction
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Entrywise nonnegative (entrywise ) nonnengative means different from positive semidefinite
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Strictly positive strictly positive means different from positive definite
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Remark e.g. nonzero, nonnegative but not positive semipositive≡nonzero, nonnegative
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Remark e.g.
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2-2 Perron’s Theorem
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spectral radius spectral radius 譜半徑
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Example
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Proven in next page
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Collatz Wielandt collatz weilandt
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Lemma 2.2.2 (1) Proven in next page
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Lemma 2.2.2 (2) Proven in next page ( 證明很重要 ) is closed and bounded above
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Remark Proven in next page
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generalized eigenvector u is called generalized eigenvector ofA if
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Remark Proven in next page the columns of P are the generalized eigenvectors of A.
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Remark The geometric multiple of λ =1 and there is no generalized eigenvector other than eigenvector corr. to λ
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Remark Proven in next page
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Remark Proven in next page
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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 )
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Theorem 2.2.1 p.1 (Perron’s Thm) (b) (c) (a)
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(f) (g) (e) (d) A has no nonnegative eigenvector other than (multiples of) u.
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Norm on a vector space (i) (iii) (ii) is a norm on V = hold iff x=0
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we introduce a metric is a metric space with on V, by
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Convergent matrix sequence can be interpreted in where one of the following equivalent way: (i)
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is in any fixed norm of where The topology of (ii) is independent of
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(the maximum norm) to be we obtain (i) In (ii), take
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Bounded matrix sequence (ii) (i) is bounded means
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Fact 2.2.4 (ii) (i)
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(iii)
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Apply of Fact 2.2.4 (ii) and P is nonsigular If then convergent problem of A is corresponding to convergent problem of
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Theorem 2.2.3 Let (i) The sequence converges to the zero matrix iff
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(ii) converges iff or and 1 is the only eigenvalue with modulus 1 and the corresp. Jordan blocks are all
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(iii) is bounded iff either and ifthen or
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Lemma 2.2.5 (i) If (ii) If then and m=1, then
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(iii) If the sequence and m=1, then is bounded Note: In this case, the seqence does not converge if explain in next page
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θ
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(iv) If then the sequence or is unbounded and
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Exercise 2.2.7 eigenvalue and is non-nipotent for every Suppose that is a simple
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what can you tell about the vector Prove that x and y? exists and is of the form
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2-3 Nonnegative Matrices
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Lemma 2.2.2 is closed, bounded above and If, then
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Lemma 2.3.1 If, then
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Lemma 2.3.2 If, then
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Fact
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Corollary 2.3.3, and B is a principal submatrix of A If then In particular
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Exercise 2.3.4 then If Hint: There is some α>1 such that
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Theorem 2.3.5 (Perron-Frobenius Thm), thenIf and
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R i (A) = i th row sum of A
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C j (A) = j th column sum of A
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Corollary 2.3.6 Then Let and
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Matrix norm is called a matrix norm if N( - ) is a A norm N( - ) on norm on, and N( - ) is submultiplicative i.e.
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Matrix norm Induced by Vector norm be a (vector) norm on Let Define onby matrix norm induced by the vector norm
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Proposition of matrix norm induced by vector norm
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Remark 2.3.7 is a marix norm on If then
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not Euclidean matrix norm correct proof in next page ( 很重要 )
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Special norm:l ∞, l p
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Special Matrix norm be the matrix norm on Let induced by the norm of
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Corollary If the row sums of A are constant Let then A row sum of A
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Exercise 2.3.8 p.1 max absolute column sum of A
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Exercise 2.3.8 p.2 max absolute row sum of A
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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 ]
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Remark 2.3.10 If A has equal row sums, then Let If A has equal column sums, then
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a row stochastic matrix with row sums all equal to 1,then A is called a row stochastic matrix. If
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a column stochastic matrix with column sums all equal to 1,then A is called a column stochastic matrix. If
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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
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Diagonally Similar p.1 are diagonal similar In particular if there is nonsingular matrix D s.t.
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Diagonally Similar p.2 preserves the class of nonnegative (as well as, positive) matrices. In particular nonnegative diagonal similarity
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Corollary 2.3.12 Then for any positive vector and we have
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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
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Exercise 2.3.13 p.2 (we adopt the convention that inf ψ=∞) Prove that for any semipositive x, we have
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Exercise 2.3.13
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Exercise 2.3.14 p.1 (i) Prove that if Let for some positive vector x then
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Exercise 2.3.14 p.2 (ii) Prove that if Let for some positive vector x then
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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
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Exercise 2.3.15 p.1 are diagonally similar. Show that the matrices and
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Exercise 2.3.15 p.2 are diagonally similar ? Are the matrices and
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