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Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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How do you write $A A^T$ in Einstein notation?

In index notation it makes sense as $$\sum_j {A_{ij} A_{jk}^T} = \sum_j {A_{ij} A_{kj}}.\tag{1}$$ But this doesn't make sense for Einstein notation where in $$A^\mu_\sigma (A^\sigma_\nu)^T = A^\m …
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