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ON THE EXISTENCE OF NOWHERE-ZERO VECTORS FOR LINEAR TRANSFORMATIONS

Part of: Basic linear algebra

Published online by Cambridge University Press:  14 September 2010

S. AKBARI ,
K. HASSANI MONFARED ,
M. JAMAALI ,
E. KHANMOHAMMADI  and
D. KIANI
S. AKBARI*
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, PO Box 11155-9415, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran (email: s_akbari@sharif.edu)
K. HASSANI MONFARED
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), PO Box 15875-4413, Tehran, Iran (email: k1monfared@gmail.com)
M. JAMAALI
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, PO Box 11155-9415, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran (email: jamaali@mehr.sharif.edu)
E. KHANMOHAMMADI
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), PO Box 15875-4413, Tehran, Iran (email: ehssanlink@gmail.com)
D. KIANI
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), PO Box 15875-4413, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran (email: dkiani@aut.ac.ir)
*
✉For correspondence; e-mail: s_akbari@sharif.edu
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Abstract

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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.

Keywords

Alon–Jaeger–Tarsi (AJT) conjecturenowhere-zero vectorLU decomposition

MSC classification

Secondary: 15A03: Vector spaces, linear dependence, rank 15A23: Factorization of matrices
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).

References

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