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Nonstandard rank-one nonincreasing maps on symmetric matrices
- Source :
- Linear and Multilinear Algebra, pp. 1-42, 2018.
- Publication Year :
- 2018
- Publisher :
- Taylor & Francis, 2018.
-
Abstract
- A map that is defined on the set of all symmetric matrices over a field 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 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.
- Subjects :
- Multilinear map
Pure mathematics
Algebra and Number Theory
Rank (linear algebra)
homogeneous polynomials
preserver problems
Field (mathematics)
010103 numerical & computational mathematics
01 natural sciences
quadratic forms
Interpretation (model theory)
Set (abstract data type)
Finite field
udc:514
symmetric matrices
symmetric tensors
Symmetric matrix
0101 mathematics
finite fields
Mathematics
Vector space
Subjects
Details
- Language :
- English
- ISSN :
- 03081087
- Database :
- OpenAIRE
- Journal :
- Linear and Multilinear Algebra, pp. 1-42, 2018.
- Accession number :
- edsair.doi.dedup.....904ddec482fe927b7c34ae09ceb06c87