On the Existence of Nowhere-Zero Vectors for Linear Transformations

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 AJT n (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) AJT n (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) AJT n (q) forms a semigroup.2000 Mathematics subject classification: primary 15A03; secondary 15A04, 15A23.