How can I express this linear algebra sum of outer products in tensor notation?

Question

I am curious about tensors and tensor notation and how it translates to common linear algebra stuff that I already know. For instance, we can express an outer product $AA^\top$ as a sum of outer products like so, with $j$ representing the column indices.

$$ AA^\top = \sum_j a_ja_j^\top $$

How would this be expressed in tensor notation? It would seem that we would need to concatenate all of the tensor products along a new dimension and then multiply a vector of ones along the new dimension $i$ like so,

$$ B_{ijk} = [a_1 \otimes a_1, \dots, a_n \otimes a_n]^\top $$

and then take a vector of ones as a covector and multiply it so that it sums along the $i$ dimension.

$$ \mathbb{1}^iB_{ijk} = AA^\top $$

Questions

1. Is this correct?
2. Is there a better way to express this?
3. Are there any good books or resources to recommend to go deeper into learning this sort of thing?

Thanks

Answer 1

Preliminaries.

Matrix multiplication can be seen as a contraction of a tensor product: $$(AB)^i{}_j=A^i{}_kB^k{}_j=(A\otimes B)^{ik}{}_{kj}$$

If $\mathbf A$ is $(0,2)$ tensor its transpose has entries $$(A^\intercal )_{ij}=A_{ji}$$

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Putting it together.

$$(AA^\intercal)^i{}_j=A^{ik}(A^\intercal)_{kj}=A^{ik}A_{jk}$$ Simple as that.