I've searched on Google but I could not find an example of a symmetric tensor. I've found this blog post but I cannot construct any example of a symmetric tensor. I know that a tensor $T$ is symmetric iff $T= \operatorname{Sym} T$. Could you give the simplest example, say on $\mathbb{R}^n$?
Symmetric $k$-tensor
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differential-geometry
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1In the blog post you mentioned there is actually an example of symmetric tensor at the end. – 2011-12-25
1 Answers
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Any symmetric matrix $M$ over $\mathbb{R}^n$ defines a symmetric tensor of rank two which maps ${\mathbb{R}^n}^*\times \mathbb{R}^n\rightarrow \mathbb{R}$ by $(\vec{x}^T,\vec{y})\mapsto \vec{x}^TM\vec{y}$. You may recognize this as an inner product on $\mathbb{R}^n$. The simplest example is given by the identity $I_n$, which defines the typical dot product on $\mathbb{R}^n$.
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0An inner product should be non-degenerate – 2011-12-25