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On the existence of nowhere-zero vectors for linear transformations

Akbari, S ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1017/S0004972710001619
  3. Abstract:
  4. A matrix A over a field F is said to be an AJT matrix if there exists a vector x over F such that both x and Ax have no zero component. The Alon-Jaeger-Tarsi (AJT) conjecture states that if F is a finite field, with |F|≥4, and A is an element of GL n (F), then A is an AJT matrix. In this paper we prove that every nonzero matrix over a field F, with |F|≥3, is similar to an AJT matrix. Let AJTn (q) denote the set of n×n, invertible, AJT matrices over a field with q elements. It is shown that the following are equivalent for q≥3 : (i) AJTn (q)=GL n (q) ; (ii) every 2n×n matrix of the form (A{divides}B)t has a nowhere-zero vector in its image, where A,B are n×n, invertible, upper and lower triangular matrices, respectively; and (iii) AJTn (q) forms a semigroup
  5. Keywords:
  7. Source: Bulletin of the Australian Mathematical Society ; 2010 , Pages 1-8 ; 00049727 (ISSN)
  8. URL: https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/div-classtitleon-the-existence-of-nowhere-zero-vectors-for-linear-transformationsdiv/FFA3A1AB10AC912D06BDDCCB9B4A754D