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ON THE EXISTENCE OF NOWHERE-ZERO VECTORS FOR LINEAR TRANSFORMATIONS
Published online by Cambridge University Press: 14 September 2010
Abstract
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∣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.
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 82 , Issue 3 , December 2010 , pp. 480 - 487
- Copyright
- Copyright © Australian Mathematical Publishing Association Inc. 2010
Footnotes
The research of the first author was in part supported by a grant from IPM (No. 88050212). The research of the fifth author was in part supported by a grant from IPM (No. 88050116).