H. Wielandt. Unzerlegbare, nicht negative Matrizen. Math. Zeitschift, 52:642-- 648, 1950. 125  Home/Search   Document Not in Database   Summary   Related Articles   Check

....if w contains an RL substring, then the forbidden sequence that appears next to last in the list (2) above would occur, which completes the proof of the claim, and of the lemma. For our next result we will use the following lemma about matrices with nonnegative entries, which is due to Wielandt [2], see also Gantmacher [1] We state somewhat less than Wielandt actually proved, but this form will suffice for our purposes. Lemma. H. Wielandt) If A and C are two n square matrices, and A has positive entries, suppose that jc i;j j a i;j for all 1 i; j n. If fl is any eigenvalue of C, and ....

H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642-648. 7

.... Gamma v 2 and l(G) t, a contradiction. Therefore there are relationships between l(G) and independent sets. Corollary 1 and Corollary 2 are two examples. For all primitive digraphs G with n vertices, clearly l(G) exp(G) n 2 Gamma 2n 2; where the latter inequality is the Wielandt bound [4]. M. Lewin observed in [1] that exp(G) n 2 Gamma 2n 2 implies l(G) 1. He also commented that there was some reason to expect that this upper bound on l(G) could be lowered considerably. In this paper, some upper bounds for l(G) in terms of the order and girth of G are found. Also it is ....

H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Zeit. 52:642-645(1950).

....of the cycles in G is one. Another equivalent formulation is given by Lewin [4] A strongly connected graph G is primitive if and only if there is an integer k 1 such that u k;k 1 Gamma v for some vertices u; v 2 V (G) Let G = V; E) be a primitive digraph on n vertices. In 1950, H. Wielandt [9] found that exp(G) n Gamma 1) 2 1 and showed there is a unique (up to isomorphism) digraph, W n , that attains this bound. The digraph W n = V; E) is defined as follows: V = fu i : 1 i ng and E = f(u i ; u i 1 ) 1 i n Gamma 1g [ f(u n Gamma1 ; u 1 ) u n ; u 1 )g. Since then much ....

H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Zeit. 52 (1950), 642-645.

....k such that u k v whenever u; v 2 V . The minimum such k is called the exponent of G, denoted exp(G) The local exponent of G at a vertex u 2 V , denoted exp(G : u) is the least integer k such that u k v for each v 2 V . Much work has been done on finding upper bounds for exp(G) see [14] and [3] for example) The diameter bound in Lemma 1 below was proved recently by Shen [9] and Neufeld [6] independently. Neufeld [6] characterized the case of equality in Lemma 1 with the following class of digraphs. Let the family FD consist of the following digraphs G = V; E) The vertex set V ....

H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Zeit., 52:642-645(1950).

....of G. The minimum such p is called the exponent of G, denoted exp(G) The local exponent of G at a vertex u 2 V , denoted exp G (u) or exp(u) if G is specified, is the least integer p such that u p v for each v 2 V . Let G = V; E) be a primitive digraph on n vertices. In 1950, H. Wielandt [8] found that exp(G) w n = n Gamma 1) 2 1 and showed there is a unique (up to isomorphism) digraph, W n , that attains this bound, where W n = V; E) is defined as follows: V = fu i : 1 i ng and E = f(u i ; u i 1 ) 1 i n Gamma 1g [ f(u n Gamma1 ; u 1 ) u n ; u 1 )g. In 1964, A. L. ....

H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Zeit., 52:642-645(1950).

....if w contains an RL substring, then the forbidden sequence that appears next to last in the list (2) above would occur, which completes the proof of the claim, and of the lemma. For our next result we will use the following lemma about matrices with nonnegative entries, which is due to Wielandt [2], see also Gantmacher [1] We state somewhat less than Wielandt actually proved, but this form will suffice for our purposes. Lemma. H. Wielandt) If A and C are two n square matrices, and A has positive entries, suppose that jc i;j j a i;j for all 1 i; j n. If fl is any eigenvalue of C, and ....

H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642-648.

....with jfflj = 1. Then the following two statements are equivalent. a) ffl is an eigenvalue of A(t 0 ) for some 0 t 0 1. b) ffl is an eigenvalue of A(t) for all 0 t 1. Proof: We only have to prove that a) implies b) and we may assume that ffl 6= 1. By a theorem of Frobenius (see Wielandt [5], page 642) the eigenvalue ffl is a primitive k th root of unity for some k 6= 1. By the same theorem there exists a permutation matrix P such that P Gamma1 A(t 0 )P = 0 B A 1 (t 0 ) 0 . 0 Am (t 0 ) 1 C A with irreducible matrices A j (t 0 ) Recall that a matrix is called ....

....for 0 t 1. In particular we have rank (A(t) Gamma E) n Gamma 1 for 0 t 1: Proof: As irreducibility depends only on the zero entries in a matrix, the first statement is obvious. It is well known that 1 is a simple eigenvalue of an irreducible stochastic matrix (see Wielandt [5]) Theorem 3.2 Let A and B be stochastic matrices of type (n; n) Suppose that lim k 1 A k exists and that A(t) is irreducible for 0 t 1. 6 a) For 0 t 1 we have P (t) lim k 1 A(t) k = 0 B v(t) v(t) 1 C A where the components of the vector v(t) are rational functions ....

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H. Wielandt, Unzerlegbare, nichtnegative Matrizen, Math.Z. 52 (1950), 642-648. 16

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H. Wielandt. Unzerlegbare, nicht negative Matrizen. Math. Zeitschift, 52:642-- 648, 1950. 125

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Wielandt, H. Unzerlegbare, nicht negative Matrizen. Math Z, No. 52, pp. 642-648.

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