# Summation substitution with multiple index

let's imagine to have the following quadratic form

$$\sum_{i,j} u_i^T F_i(u_i)^T F_j(u_j) u_j$$

where $$F_k$$ is a linear operator acting on the generic vector $$u_k$$

If I have the following expression for the $$u_i$$

$$u_i = v + \sum_{jk}a_{ij}c_k F_j(v)v$$

and consequently

$$F_i(u_i) = v + \sum_{jk}a_{ij}c_k F_i(F_j(v)v)$$

Now, I would like to substitute these expressions, into the first one, but I'm getting confused by the indexing. For example, if I change the index of $$u_j$$ as follows

$$u_j = v + \sum_{lm}a_{jl}c_m F_l(v)v$$

is it correct to put this expressions into the first summation ? and If I do the same for $$F_i$$, $$F_j$$ and $$u_i$$, should I keep the same index notation, or should I use different indexing for every expression I'm going to substitute into ?

For example, let's define

$$u_j = v + \sum_{lm}a_{jl}c_m F_l(v)v = h(v,j,l,m)$$

and

$$F_j(u_j) = v + \sum_{lm}a_{jl}c_m F_j(F_l(v)v)$$

$$u_j = h(v,j,l,m)$$

$$u_i = h(v,i,l,m)$$

$$F(u_j) = F(h(v,j,l,m))$$

$$F(u_i) = F(h(v,i,l,m))$$

so that

$$\sum_{i,j} u_i^T F_i(u_i)^T F_j(u_j) u_j = \sum_{i,j} h(v,i,l,m)^T F(h(v,j,l,m)) ^T F(h(v,j,l,m)) h(v,j,l,m)$$

Or should I use 4 different l and same for m for each $$u_i$$, $$F_i(u_i)$$ ??

• "$u_i = v + \sum_{ijk}a_{ij}c_k F_j(v)v$" is an abuse of notation. The first $i$ should not be the same as $i$ in the summation. Try to use different variable names. Apr 16, 2023 at 20:18
• Sorry, my bad, the summations are not over the index of the variable, of course, so it can be indeed of index i. I fixed the notation, now it should be fine Apr 16, 2023 at 21:36

The variables declared to loop through a summation are what's called dummy variables. They can be replaced by any variables that are not previously defined. They are there for notational convenience. For example, the expression $$\sum_{i=1}^n ix^2$$ only depends on the variables $$n$$ and $$x$$, but it does not depend on $$i$$. For example, you can change $$i$$ to $$j$$ or anything you like (as long as they don't mean something else) and still get the same thing.
In your case, $$u_i=v+\sum_{jk}a_{ij}c_k F_j(v)(v)$$ only depends on $$i$$, $$v$$, $$(p,q)\mapsto a_{pq}$$, $$p\mapsto c_p$$, $$p\mapsto F_p$$. You seem to imply that the maps $$(p,q)\mapsto a_{pq}$$, $$p\mapsto c_p$$, $$p\mapsto F_p$$ are all fixed, so $$u_i$$ only depends on $$i$$ and $$v$$. Thus, you should really name your function $$h$$ by $$h(i,v)$$.
Then you can safely substitute $$u_i$$ for $$h(i,v)$$ and $$u_j$$ for $$h(j,v)$$. The resultant summation does not depend on $$l,m$$ at all. It does not depend on $$i,j$$ as well, because $$i,j$$ are also variables used to loop through the summation. You may replace it by $$p,q$$ for example.
• Thank you so much, I perfectly find myself in your answer, but I hadn't been specific maybe, so I'm sorry. The goal is to obtain a final relation of the summation with the only dependence of $v$ and not $u_k$. So that I'm forced to manually (or I think to do so) the summation of the generic $u_i$ into the summation, I'm quite sure that to do that I should use different secondary maps fixed, i.e. I would use $(p,q)$ for the first $u_i$, $(t,r)$ for $F_i(u_i)$ and so on. Is it right ? Apr 17, 2023 at 8:06
• If you replace all the $u_i$ and $u_j$ by $h(i,v)$ and $h(j,v)$, then the final summation only depends on $v$ because $i$ and $j$ in the final summation are dummy variables. Apr 17, 2023 at 8:18