English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field$\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space$\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S}\rightarrow V$ is called range-compatible when it satisfies $F(s) \in\mathrm{im}(s)$ for all $s \in \mathcal{S}$. Among the range-compatible mapsare the so-called local ones, that is the maps of the form $s \mapsto s(x)$ fora fixed vector $x$ of $U$.  In recent works, we have classified the range-compatible group homomorphismson $\mathcal{S}$ when the codimension of $\mathcal{S}$ in $\mathcal{L}(U,V)$ issmall. In the present article, we study the special case when $\mathcal{S}$ isa linear subspace of the space $S_n(\mathbb{K})$ of all $n$ by $n$ symmetricmatrices: we prove that if the codimension of $\mathcal{S}$ in$S_n(\mathbb{K})$ is less than or equal to $n-2$, then every range-compatiblehomomorphism on $\mathcal{S}$ is local provided that $\mathbb{K}$ does not havecharacteristic $2$. With the same assumption on the codimension of$\mathcal{S}$, we also classify the range-compatible homomorphisms on$\mathcal{S}$ when $\mathbb{K}$ has characteristic $2$. Finally, we prove thatif $\mathcal{S}$ is a linear subspace of the space $A_n(\mathbb{K})$ of all $n$by $n$ alternating matrices with entries in $\mathbb{K}$, and the codimensionof $\mathcal{S}$ is less than or equal to $n-3$, then every range-compatiblehomomorphism on $\mathcal{S}$ is local.

Range-compatible homomorphisms on spaces of symmetric or alternating matrices