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A 4th dimensional 2nd order tensor is given as $\overline {\overline T}=2\vec e_1\vec f_1+2\vec e_2\vec f_2+\vec e_1\vec f_2+\vec e_2\vec f_1+\epsilon:\vec e_1\vec e_2$ where $\vec e_i.\vec f_j=\delta_{ij}$ and $\epsilon$ is 4-D permutation tensor. Also $\vec f_3=2\vec e_3-\vec e_4$ and $\vec f_4=\vec e_4-\vec e_3$ and $\epsilon :\vec e_1 \vec e_2\vec e_3\vec e_4=5$. How I can define the tensor T so I can find its determinant and eigenvalues?

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