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If you consider a second rank tensor as the tensor product of two four-vectors then examining how this tensor behaves under lorentz transformations is a question of multiplying all possible products of components of vectors that have gone under a lorentz transformation, which is fine but...

How do I phrase this question in terms of bilinear functionals & secondly in terms of their matrix representation? In other words, considering the Lorentz transformation as a linear transformation how would I plug a second rank tensor into this to show how it behaves under lorentz transformations?

I vaguely recall seeing this question being phrased in terms of matrix multiplication, but I don't see how this works & so I think starting from the consideration of how tensors behave under linear transformations should illuminate everything (hopefully) if such a question makes sense, thanks.

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If you have two vectors $a, b$ and a linear map $T$, then it is natural to consider the quantity $a \otimes b \mapsto T(a) \otimes T(b)$. In index notation, $a^i b^j \mapsto T_{ik} a^i T_{jl} b^j$, say. This should capture all the relevant information that you might want or need.

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