Range-compatible homomorphisms on spaces of symmetric or alternating matrices

Range-compatible homomorphisms on spaces of symmetric or alternating matrices Let U and V be finite-dimensional vector spaces over an arbitrary field K, and S be a linear subspace of the space L(U,V) of all linear maps from U to V. A map F:S→V is called range-compatible when it satisfies F(s)∈Ims for all s∈S. Among the range-compatible maps are the so-called local ones, that is the maps of the form s↦s(x) for a fixed vector x of U.In recent works, we have classified the range-compatible group homomorphisms on S when the codimension of S in L(U,V) is small. In the present article, we study the special case when S is a linear subspace of the space Sn(K) of all n by n symmetric matrices: we prove that if the codimension of S in Sn(K) is less than or equal to n−2, then every range-compatible homomorphism on S is local provided that K does not have characteristic 2. With the same assumption on the codimension of S, we also classify the range-compatible homomorphisms on S when K has characteristic 2. Finally, we prove that if S is a linear subspace of the space An(K) of all n by n alternating matrices with entries in K, and the codimension of S is less than or equal to n−3, then every range-compatible homomorphism on S is local. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Linear Algebra and its Applications Elsevier

Range-compatible homomorphisms on spaces of symmetric or alternating matrices

Loading next page...
 
/lp/elsevier/range-compatible-homomorphisms-on-spaces-of-symmetric-or-alternating-6FXujp8fyr
Publisher
Elsevier
Copyright
Copyright © 2016 Elsevier Inc.
ISSN
0024-3795
eISSN
1873-1856
D.O.I.
10.1016/j.laa.2016.03.047
Publisher site
See Article on Publisher Site

Abstract

Let U and V be finite-dimensional vector spaces over an arbitrary field K, and S be a linear subspace of the space L(U,V) of all linear maps from U to V. A map F:S→V is called range-compatible when it satisfies F(s)∈Ims for all s∈S. Among the range-compatible maps are the so-called local ones, that is the maps of the form s↦s(x) for a fixed vector x of U.In recent works, we have classified the range-compatible group homomorphisms on S when the codimension of S in L(U,V) is small. In the present article, we study the special case when S is a linear subspace of the space Sn(K) of all n by n symmetric matrices: we prove that if the codimension of S in Sn(K) is less than or equal to n−2, then every range-compatible homomorphism on S is local provided that K does not have characteristic 2. With the same assumption on the codimension of S, we also classify the range-compatible homomorphisms on S when K has characteristic 2. Finally, we prove that if S is a linear subspace of the space An(K) of all n by n alternating matrices with entries in K, and the codimension of S is less than or equal to n−3, then every range-compatible homomorphism on S is local.

Journal

Linear Algebra and its ApplicationsElsevier

Published: Aug 15, 2016

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off