Nonstandard rank-one nonincreasing maps on symmetric matrices

Povzetek

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.

Ključne besede

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

Podatki

Jezik:	Angleški jezik
Leto izida:	2019
Tipologija:	1.01 - Izvirni znanstveni članek
Organizacija:	UP - Univerza na Primorskem
UDK:	512.64
COBISS:	1540253892
ISSN:	0308-1087
Št. ogledov:	2746
Št. prenosov:	178
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Sekundarni jezik:	Angleški jezik
Vrsta dela (COBISS):	Članek v reviji
Strani:	str. 391-432
Letnik:	ǂVol. ǂ67
Zvezek:	ǂiss. ǂ2
Čas izdaje:	2019
DOI:	10.1080/03081087.2017.1419456
ID:	10927222