I have read in a couple of places that mixed tensors cannot be decomposed into a sum of symmetric and antisymmetric parts. This doesn't make any sense to me because I thought a mixed (1,1) tensor was basically equivalent to a standard linear transform from basic linear algebra. I am also having trouble with finding anything that shows how changing the order of the indexes effects a mixed tensor and how basic coordinate transforms affect mixed tensors with permuted indexes. The background is basic relativity so I usually use lorentz transforms and minkowski metric for examples. Here is my idea of the symmetric part of a mixed tensor M.(adsbygoogle = window.adsbygoogle || []).push({});

[itex] \frac{1}{2}\left[ {M^\alpha}_\beta + {N^\alpha}_\beta \right] [/itex]

where (i think)

[itex] {N^\alpha}_\beta = g_{\nu\beta}{M^\nu}_\mu g^{\mu\alpha} = {M_\beta}^\alpha[/itex]

I am having trouble seeing how to transform it to a new coordinate system because every book I have (and I have about 10) shows the coordinate transform matrices and mixed tensors with upper and lower indices directly above/below each other, so I am not sure how they apply.

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# Symmetric Part of a Mixed (1,1) Tensor

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