For vector $y=[y_1,y_2,\dots y_n]$ , let $\gamma = \sum_{i=1}^n \gamma_i$ , and $\gamma_i(n-i+1)=y_n*y_{n-1}*\dots y_i$ so that $\gamma$ looks like $[y_1*y_2*\dots y_n, y_2*\dots*y_{n}, \dots y_n]$ composed of $\gamma_i$`s that look like $\gamma_1=[0,0, \dots y_n]$ and $\gamma_2=[0,0, \dots y_{n-1}y_{n},0]$ etc. How do I generalise a linear algebra function that creates $\gamma$ from $y$? We have solved this up to n=3 in another question here. Those solutions (for $y=[a,b,c]$ and $\gamma=[abc,bc,c]$ so $n=3$) looked like this:
$$y=[a,b,c]$$
$$\gamma_{n-1+1}=\gamma_3=(Iy)\pmatrix{0\cr0\cr1}=[0,0,c]$$
$$\gamma_{n-2+1}=\gamma_2=(I[(y^Ty)\pmatrix{0\cr0\cr1}])\pmatrix{0\cr1\cr0}=[0,bc,0]$$
$$\gamma_{n-3+1}=\gamma_1=(I[y\pmatrix{0,0,0\cr0,0,1\cr0,0,0}(y^Ty)])\pmatrix{1\cr0\cr0}=[abc,0,0]$$
$$\gamma=\sum_{i=1}^3\gamma_i=[abc,bc,c]$$
$$y=[y_1,\dots ,y_n] \to ?\to \gamma=[y_ny_{n-1}y_{n-2}\dots y_1, y_ny_{n-1}y_{n-2}\dots y_2,\dots ,y_n]$$
My question is how to formulate a linear algebra sequence for $\gamma_i$ like above, but then generalized so you can apply it to $n$ entries of the $n$ vector $y$.
I am taking my first linear algebra course, so its quite possible that I am doing something that has a correct name for it, but in that case I dont know what it is that I am doing. Please let me know any comments on my question and I will edit and I will answer your questions as well. Thanks for any help.
To explain what I do above: the $y^Ty$`s create a matrix with multiplications of the terms of $y$ to the degree of $n$ by multiplying again with $y$, $n$ times. Then I select the column or row that has the right multiplication in it ($abc$ for $n=3$, $bc$ for $n=2$ and $c$ for $n=1$) then multiply that with $I$ and then multiply that with a base vector to create $\gamma_i$. So if $y=[1,2,\dots ,n]$ then $\gamma=[n!,\frac{n!}{[n-(n-1)]!},\frac{n!}{[n-(n-2)]!},\dots ,n]$. The brackets in my answer for $n=3$ show the order in which I did it manually, but according so associativity they dont really matter I guess.