Capital Pi notation inverse direction

Question

I was kind of surprised that I could not find the answer while searching for it here, the question is really simple I have a series of transformations, denoted by a 2x2 matrix $c_i$. To come from the from the initial to the final state on has to calculate

$$X'=c_nc_{n+1}...c_{2}c_iX$$

I would like to write this in capital Pi notation in the simplest and most understandable way, but I am not sure what the best way is since the order is important. My guess would be:

$$X'=\left(\prod_{i=n}^1c_1\right)X$$

I am wondering if this is both, correct and understandable.

Thanks

Edit: To be more clear, the standard notation $$\prod_{i=1}^nc_n = c_1...c_n$$ is not the product I want, since matrices do not commute in general

Edit: Of course $\prod_{i=1}^nc_{n+1-i}$ is a correct and easy way of writing the multiplication. I just assume one has to think briefly about the indices. The version in the unedited question is planned to say "start with n, stop with 1" which is imo even simpler.

Answer 1

I think you mean $X'=c_nc_{n-1}...c_{2}c_1X$. If this is the case you can write:

$X'=\prod_{i=0}^{n-1} c_{n-i}$.

Answer 2

The products indexed in this notation don't do what you think; the index set is oriented, so iterating backwards has the opposite effect of iterating forwards.

More precisely they're defined to still satisfy the usual recursion:

$$ \left(\prod_{i=a}^{b-1} s_i \right) s_b = \left( \prod_{i=a}^{b} s_i \right) $$

and more generally, the identity

$$ \left(\prod_{i=a}^{b} s_i \right) \left( \prod_{i=b+1}^c s_i \right) = \prod_{i=a}^c s_i $$

In a particular, you'd have

$$ \prod_{i=n}^1 c_i = \left( \prod_{i=2}^{n-1} c_i \right)^{-1} $$

Note, however, that actually using index sets that run backwards is uncommon enough that you shouldn't rely on the audience being familiar; if you were to actually use them you should explain the meaning.

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(as suggested by the recursion above, I believe the usual convention on forward products that the individual factors are listed from left to right)