---- > [!definition] Definition. ([[symmetric multilinear map]]) > Let $R$ be a [[ring]] and let $M_{1},\dots,M_{\ell}, P$ be $R$-[[module|modules]]. An $R$-[[multilinear map|multilinear map]] $\varphi: M_{1} \times \dots \times M_{\ell} \to P$ is called **symmetric** if it is [[group-invariant function|invariant]] under [[group action|actions]] of the [[symmetric group]] $S_{\ell}$, i.e., if for all [[permutation|permutations]] $\sigma \in S_{\ell}$ and elements $(m_{1},\dots,m_{\ell}) \in M_{1} \times \dots \times M_{\ell}$ we have $\varphi(m_{\sigma(1)}, \dots, m_{\sigma(\ell)})=\varphi(m_{1},\dots,m_{\ell}).$ > In the special case $\ell=2$ and $P=R$, $\varphi$ is a **symmetric bilinear form**. Evidently, [[matrix of a bilinear form|the matrix of]] $\varphi$ [[matrix of a bilinear form|is]] [[symmetric matrix|symmetric]] wrt any choice of [[basis]]. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```