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I would like to create symbolic tensor. For example a 2nd order symmetric tensor that would look like this in matrix representation:

$$ \left( \begin{array}{ccc} a & d & f \\ d & b & e \\ f & e & c \\ \end{array} \right) $$

Is this possible for higher order tensors (order 4, for example)? I don't feel like typing out all 81 components. And more importantly, will further calculations with these tensors simplify due to symmetry (if applicable)?

My goal is to find zero elements of the following tensor: : $$ (c_{n_1 m_1}+c_{n_2m_2}+...+c_{n{_\mu m_{\mu}}})D_{m_1 m_2 ... m_{\mu}} $$

($D_{m_1 m_2 ... m_{\mu}}$ symmetric in all its indices, $c_{ij}$ is antisymmetric and zero everywhere except $c_{12}=\theta$ and $c_{21}=-\theta$, working in 3 dimensions)

I would like to create symbolic tensor. For example a 2nd order symmetric tensor that would look like this in matrix representation:

$$ \left( \begin{array}{ccc} a & d & f \\ d & b & e \\ f & e & c \\ \end{array} \right) $$

Is this possible for higher order tensors (order 4, for example)? I don't feel like typing out all 81 components. And more importantly, will further calculations with these tensors simplify due to symmetry (if applicable)?

My goal is to find zero elements of the following tensor: : $$ (c_{n_1 m_1}+c_{n_2m_2}+...+c_{n{_\mu m_{\mu}}})D_{m_1 m_2 ... m_{\mu}} $$

($D_{m_1 m_2 ... m_{\mu}}$ symmetric in all its indices, $c_{ij}$ is antisymmetric and zero everywhere except $c_{12}=\theta$ and $c_{21}=-\theta$, working in 3 dimensions)

I would like to create symbolic tensor. For example a 2nd order symmetric tensor that would look like this in matrix representation:

$$ \left( \begin{array}{ccc} a & d & f \\ d & b & e \\ f & e & c \\ \end{array} \right) $$

Is this possible for higher order tensors (order 4, for example)? And more importantly, will further calculations with these tensors simplify due to symmetry (if applicable)?

My goal is to find zero elements of the following tensor: : $$ (c_{n_1 m_1}+c_{n_2m_2}+...+c_{n{_\mu m_{\mu}}})D_{m_1 m_2 ... m_{\mu}} $$

($D_{m_1 m_2 ... m_{\mu}}$ symmetric in all its indices, $c_{ij}$ is antisymmetric and zero everywhere except $c_{12}=\theta$ and $c_{21}=-\theta$)

I would like to create symbolic tensor. For example a 2nd order symmetric tensor that would look like this in matrix representation:

$$ \left( \begin{array}{ccc} a & d & f \\ d & b & e \\ f & e & c \\ \end{array} \right) $$

Is this possible for higher order tensors (order 4, for example)? I don't feel like typing out all 81 components. And more importantly, will further calculations with these tensors simplify due to symmetry (if applicable)?

My goal is to find zero elements of the following tensor: : $$ (c_{n_1 m_1}+c_{n_2m_2}+...+c_{n{_\mu m_{\mu}}})D_{m_1 m_2 ... m_{\mu}} $$

($D_{m_1 m_2 ... m_{\mu}}$ symmetric in all its indices, $c_{ij}$ is antisymmetric and zero everywhere except $c_{12}=\theta$ and $c_{21}=-\theta$, working in 3 dimensions)

I would like to create symbolic tensor. For example a 2nd order symmetric tensor that would look like this in matrix representation:

$$ \left( \begin{array}{ccc} a & d & f \\ d & b & e \\ f & e & c \\ \end{array} \right) $$

Is this possible for higher order tensors (order 4, for example)? And more importantly, will further calculations with these tensors simplify due to symmetry (if applicable)?

My goal is to find zero elements of the resulting tensor to solve a tensor equation like this ($D$ symmetric in all its elements, $c_{ij}$ is explicitly known): $$ (c_{n_1 m_1}+c_{n_2m_2}+...+c_{n{_\mu m_{\mu}}})D_{m_1 m_2 ... m_{\mu}}=0 $$

My idea was to calculate each of the partial products with TensorProduct[], sum them and that's it. I just need to know which components are zero. As of now, Mathematica is able to find some explicit form, but it does not simplify the result.

I would like to create symbolic tensor. For example a 2nd order symmetric tensor that would look like this in matrix representation:

$$ \left( \begin{array}{ccc} a & d & f \\ d & b & e \\ f & e & c \\ \end{array} \right) $$

Is this possible for higher order tensors (order 4, for example)? And more importantly, will further calculations with these tensors simplify due to symmetry (if applicable)?

My goal is to find zero elements of the following tensor: : $$ (c_{n_1 m_1}+c_{n_2m_2}+...+c_{n{_\mu m_{\mu}}})D_{m_1 m_2 ... m_{\mu}} $$

($D_{m_1 m_2 ... m_{\mu}}$ symmetric in all its indices, $c_{ij}$ is antisymmetric and zero everywhere except $c_{12}=\theta$ and $c_{21}=-\theta$)