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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 $u_i$, $F_i(u_i)$ ??

Thanks so much in advance.

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  • $\begingroup$ "$u_i = v + \sum_{ijk}a_{ij}c_k F_j(v)v$" is an abuse of notation. The first $i$ should not be the same as $i$ in the summation. Try to use different variable names. $\endgroup$
    – durianice
    Apr 16, 2023 at 20:18
  • $\begingroup$ 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 $\endgroup$
    – Marco
    Apr 16, 2023 at 21:36

1 Answer 1

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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.

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  • $\begingroup$ 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 ? $\endgroup$
    – Marco
    Apr 17, 2023 at 8:06
  • $\begingroup$ 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. $\endgroup$
    – durianice
    Apr 17, 2023 at 8:18

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