Why is the ray transfer matrix system products from right to left? Example: link


*

*Process 1: T1 (Start)

*Process 2: S1 

*Process 3: T2 

*Process 4: S2 

*Process 5: T3 (End)


System matrix = product of all process = T3.S2.T2.S1.T1
Why does it start from T3 to T1 and not from T1 to T3?
 A: This is simply owing to the way we define the action of transfer matrices (not just ray transfer ones) in general. 
We could instead define transfer matrices to act from the right on row vectors instead of from the left on column vectors. If that's a little confusing, it's best illustrated by example. You can easily check that:
$$\left(\begin{array}{c}x\\y\end{array}\right) \mapsto \left(\begin{array}{cc}a & b\\c & d\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)$$
is exactly the same transformation as:
$$\left(\begin{array}{cc}x&y\end{array}\right) \mapsto \left(\begin{array}{cc}x&y\end{array}\right) \left(\begin{array}{cc}a & c\\b & d\end{array}\right)$$
(note that the matrix has been transposed). In the first convention, where matrices act on column vectors, if you think of imparting transformations in a sequence of steps, you see that we chain the matrices from right to left as in your question. But in the second convention, if you impart the successive steps again, you see that we chain them from left to right!
We choose the former convention so that the composition of the matrices of transformations mimics the function / operator composition notation $F\circ G(x)\stackrel{def}{=}F(G(x))$. However, algebraicists sometimes write functions as acting on the right of their argument for the same reason as you want to: composition then chains left to right: this clarifies complicated symbolic descriptions for some people. Mathematica, for example, supports this acting on the right notation.
This practice (i.e. with matrices acting on row vectors) is probably not recommended unless you have very good reason and you state your convention clearly: you're likely to cause confusion otherwise.
