1
$\begingroup$

I have a question concerning the ordering of the index in the product symbol. Please take a look at the highlighted indices in the following image:

enter image description here

In equation (4), I read the summation as "starting" at $k = 1$, and "ending" at $k = t$.

In equation (5) however, I am reading the product as "starting" at $i = t$, and "ending" at $i = k+1$.

Have I read this correctly? Or do I have it backwards? Is this what the notation is meant to convey here?

1 Answers 1

0

For the sum the order of the finitely many summands doesn't matter, so you need not think about whether the sum starts or ends at $1$.

For the product it won't matter either if the factors are numbers. If they are matrices (operators) then it might. If it does, I think conveying that information in the product sign is not a particularly good idea.

Note that one of the index lists includes $k=1$; the other doesn't. Presumably that matters.

  • 0
    Ethan, yes the quantity inside the product is indeed a matrix. (A Jacobian), hence my elevated caution on how to open it. Let me ask this: Given that this is a matrix inside the product symbol, how would you open this up upon sight? – 2017-01-18
  • 0
    The only way to "open it up on sight" is know that the factors are matrices, so the order might matter, and then pay attention to the way it's written. You seem to have done just that. If I were writing it I might call attention to the order in the text before the displayed equation. Or write it out with the first, second and last factors separated by an ellipsis. – 2017-01-18