# Identification of matrix index notation

I came across this expression in the documentation for some software:

$\displaystyle\sum\limits_{j=1}^c\alpha_{i*(j)k}q_j=Q_k$

Here, $\alpha$ is the element of a matrix with the rather curious index "$i*(j)k$", which the documentation describes as being a mapping function from the elements of one matrix to another. I have never seen this notation before in my life, specifically the use of the multiplication operator and the parentheses. I am familiar with Einstein notation for matricies, but this does not seem to be related to that. It is possible that this is just a typo in the documentation, but in case anyone can make sense of it, please help me understand! Thanks.

• A further perplexity of that definition is the meaning of "$i$" in there. Is there context for that? Sep 25 '14 at 15:56
• The software is for analyzing chemical reactions; "$i$" in this case is the index of a chemical element that appears in the reaction. For example, $i=1$ is carbon, $i=2$ is hydrogen, etc. Sep 25 '14 at 16:03
• Gotcha. Best I can guess is that they intend it as $\alpha_{i*(j),k}$ i.e. $\alpha$ as an array. But that still leaves the $i*(j)$ mysterious. Sep 25 '14 at 16:13
• Yes, I forgot to mention that $\alpha$ is a 2D matrix/array, so it definitely needs a comma in there somewhere to denote the two indicies. I think you're correct in that there's a missing comma. Sep 25 '14 at 16:19
• I'm guessing the star is there in order to visually separate $a_i$, the array for a particular chemical element, from the indices $j$ and $k$? But this still doesn't explain the parentheses on $(j)$, so who knows... Sep 25 '14 at 16:29

As far as I can tell, this obscure notation was invented by the authors of the paper. In short, there are two vectors $I$ and $J$, and by definition $J$ is a subset of $I$. Therefore, any element $J_j$ has a corresponding element $I_i$, and the mapping function between these two elements is denoted by "$i*(j)$". In practice, this is used to say that when the index $j$ is used (such as in the summation in my original post), we can instead map it to a corresponding index $i$ and pull the value of interest from $I$ instead of $J$.