# Tensor notation for 3-D matrix expression

I have the expression

$y_i = \displaystyle\sum_j x_j \!\cdot\!a_{ij} \!\cdot\! \exp \Big(\sum_k b_{ijk} \!\cdot\! x_k \Big)$

which I want to shorten without introducing more notation than necessary. For example, $\displaystyle\,y_i = \sum_j a_{ij}\! \cdot\! x_j\,$ can be written as $\,\vec{y} = A \,\vec{x}\,$ without any further explanation.

$\vec{y} = \Big(A \!\circ\! \exp\big(\, \underline{B} \!\cdot \!\vec{x} \,\big) \Big) \!\cdot\! \vec{x}$

where $A$ is the matrix with entries $a_{ij}$ and $B$ is the 3rd order tensor-like thing with entries $b_{ijk}$? Of course, $\exp$ is taken element-wise.

I am saying tensor-like because I am not sure which notation I would have to use when I were to use tensor index notation – for example, $b^{ij}_k$ or $b^i_{jk}$. All beginner articles about tensor notation seem to use the concept of a changing basis to determine which index goes super- or subscript; however, I only have one single basis, so proceeding like this does not really help me.

Update: I think my main question is this: If $\,\underline{B}\,$ is a 3rd order tensor and $\,x\,$ a vector, is it obvious (or usually assumed) that $\,\displaystyle\underline{B} \!\cdot\!\vec{x} = \sum_k b_{ijk}\!\cdot\!x_k$? Or could this be understood as $\,\sum_j b_{ijk}\!\cdot\!x_j\,$ or even $\,\displaystyle\sum_i b_{ijk}\!\cdot\!x_i$?

Another update: Similar to the above: What about $\,\vec{x}^{\,T} \!\!\!\cdot\! \underline{B}$? I understand that as $\,\displaystyle\sum_i x_i \!\cdot\!b_{ijk}$, but is this obvious?

• I am curious where your desired expression comes from. Tensor notation is not really intended to handle element-wise operations (aside from addition and scalar multiplication, of course) and usually the exponential of a second-order tensor is defined like the matrix exponential. Oct 7, 2015 at 20:22
• The exponential comes from the Beer-Lambert law; the expression describes a physical model of a photon-propagation process.
– bers
Oct 7, 2015 at 20:23
• In a slightly different direction, the Einstein summation convention would be helpful here. As @MattDickau said, it's really only useful for 'flat' expressions, but it's helpful (and generally cleaner than the vector-and-dot-product approach). Jan 27, 2016 at 15:55
• The problem with that notation is, how to you pronounce the difference between $\exp(\sum_j a_{ij} x_j)$ and $\sum_j \exp(a_{ij} x_j)$? Both will be $\exp(a_{ij} x_j)$, I guess.
– bers
Jan 27, 2016 at 15:59
• This is one of the reasons why Einstein summation convention exists, and why past 2-D matrices, you usually write all the sub/superscripts. There are just too many different ways to multiply tensors together to have a different symbol for each case. With Einstein summation, you could write this as $x_i \cdot a_{ij} \cdot exp(b_i^{jk} \cdot x_k)$, which somewhat works. Feb 3, 2016 at 4:39

On the other hand, your own "tensor-like" notation looks very clean and intuitive, but it requires a bit more explanation in your text. This task may be facilitated by the notation being "Matlab style", in the sense that functions on arrays are, by default, evaluated element-wise and that there is an element-wise multiplication of arrays, indicated by a dot "." in Matlab and in your notation by "$\circ$". So it will appear familiar to most readers.