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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

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  • $\begingroup$ Your notation for $B_{ijk}$ doesn't make much sense. First of all, it looks like you're constructing a block matrix rather than concatenating along a new dimension, and second the right hand side makes no reference to the indices $j,k$ that are present on the left. $\endgroup$ Sep 13, 2022 at 15:17
  • $\begingroup$ Instead, I think the most sensible way to write this vector is as $$ B_{ijk} = [a_k \otimes a_k]_{ij}, $$ or more simply as $B_{ijk} = a_{ki}a_{ji}$. $\endgroup$ Sep 13, 2022 at 15:18
  • $\begingroup$ How can I read the $ij$ index in $B_{ijk} = [a_k \otimes a_k]_{ij}? I can see all the pieces are there but I can't make sense of it in my head. $\endgroup$
    – Joff
    Sep 13, 2022 at 15:31
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    $\begingroup$ By the way, I think it would be more natural to express $AA^\top$ as the contraction of the order-4 tensor product $B = A \otimes A^\top$, which is expanded in two new dimensions. $\endgroup$ Sep 13, 2022 at 15:47
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    $\begingroup$ You might find this post on the connection between tensor contraction and matrix multiplication to be useful. I don't have a recommendation, but you might find this list to be helpful $\endgroup$ Sep 13, 2022 at 16:41

1 Answer 1

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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 $$(A^\intercal )_{ij}=A_{ji}$$


Putting it together.

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

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