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 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

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
Access to DeepDyve database
Abstract access only
Unlimited access to over
18 million full-text articles
Print
20 pages/month
PDF Discount
20% off