Nonstandard rank-one nonincreasing maps on symmetric matrices

Marko Orel (Author)

Abstract

A map that is defined on the set of all ▫$n\times n$▫ symmetric matrices over a field ▫$\mathbb F$▫ is rank-one nonincreasing if it maps the matrices of rank one to matrices of rank at most one. In the case of the fields with two or three elements nonstandard examples of such additive maps are known to exist. In this paper, all these maps are characterized. Moreover, the existence of nonstandard maps is understood in a geometric sense since they exist precisely in the two cases, where the whole ▫$n$▫-dimensional vector space over ▫$\mathbb F$▫ can be written as a union of two hyperbolic/parabolic quadrics of low indices. This geometric interpretation enables us to realize that these weird examples represent just a tip of an iceberg that appears in an analogous problem on symmetric tensors, where nonstandard maps exist also over larger fields. A few open problems related to the multilinear analog and to homogeneous polynomials are posed.

Keywords

quadratic forms;symmetric matrices;preserver problems;finite fields;symmetric tensors;homogeneous polynomials;

Data

Language:	English
Year of publishing:	2019
Typology:	1.01 - Original Scientific Article
Organization:	UP - University of Primorska
UDC:	512.64
COBISS:	1540253892
ISSN:	0308-1087
Views:	2746
Downloads:	178
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Type (COBISS):	Article
Pages:	str. 391-432
Volume:	ǂVol. ǂ67
Issue:	ǂiss. ǂ2
Chronology:	2019
DOI:	10.1080/03081087.2017.1419456
ID:	10927222