Algebraic Operations with Summation within summation

Question

Given $x_i = (X_i - bar{X})$, where $bar{X}$ is the sample mean of $X$; and given the variable $k_i = frac{x_i}{sum x_i^2}$.
How do I show that $sum k_i x_i = sum k_i X_i = 1$?
I don't know how to incorporate information about the sample mean. Perhaps this has some statistics knowledge that I don't know? I don't know how to deal with a summation within a summation that all uses i.

d.k.o. · Answer

Let $X:=[X_1,ldots,X_n]^{top}$ and $P:=mathbf{i}mathbf{i}^{top}/n$, where $mathbf{i}:=[1,ldots, 1]^{top}$. Note that the matrix $I-P$ is symmetric and idempotent, i.e., $I-P=(I-P)^{top}$ and $(I-P)(I-P)=I-P$. Therefore,
$$
sum_{i=1}^n k_iX_i=frac{X^{top}(I-P)X}{((I-P)X)^{top}(I-P)X}=frac{X^{top}(I-P)X}{X^{top}(I-P)X}=1.
$$

Algebraic Operations with Summation within summation

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