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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 $\mathbb{R}^n$?

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In the blog post you mentioned there is actually an example of symmetric tensor at the end. –  Soarer Dec 25 '11 at 7:44
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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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An inner product should be non-degenerate –  Yuri Vyatkin Dec 25 '11 at 23:13
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