Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T04:52:03.738Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
K. HASSANI MONFARED
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), PO Box 15875-4413, Tehran, Iran (email: [email protected])
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: [email protected])
E. KHANMOHAMMADI
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), PO Box 15875-4413, Tehran, Iran (email: [email protected])
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: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Akbari, S., Dorbidi, H. R. and Jamaali, M., ‘A variation of Alon–Jaeger–Tarsi conjecture’, submitted.Google Scholar
[2]Alon, N. and Tarsi, M., ‘A nowhere-zero point in linear mappings’, Combinatorica 9(4) (1989), 393–395.CrossRefGoogle Scholar
[3]Baker, R. D., Bonin, J., Lazebnik, F. and Shustin, E., ‘On the number of nowhere zero points in linear mappings’, Combinatorica 14(2) (1994), 149–157.CrossRefGoogle Scholar
[4]Horn, R. A. and Johnson, C. R., Matrix Analysis (Cambridge University Press, Cambridge, 1985).CrossRefGoogle Scholar
[5]Kirkup, G. A., ‘Minimal primes over permanental ideals’, Trans. Amer. Math. Soc. 360(7) (2008), 3751–3770.CrossRefGoogle Scholar