0
$\begingroup$

Given $d$-dimensional coordinates residing in a matrix $X\in\mathbb{R}^{n\times d}$, the Euclidean distance between items $i$ and $j$ is denoted as $g_{ij}$. Let $c\in\mathbb{R}^d$ denote the centroid of the configuration $X$. Is it possible to establish a relation between $$\sum_{i,\, j} g_{ij}^2,$$ and $$\sum_{i=1}^n g_{ic}^2,$$where $g^2$ is a squared Euclidean distance? My solution shows that the first is $2n$ times the second. Is this correct?

$\endgroup$
1
$\begingroup$

Yes. Assume wlog that the centroid is at zero. Then $\sum_i x_i=0$, whence $\sum_{i,j} g_{ij}^2 = \sum_{i,j}\|x_i-x_j\|^2=\sum_{i,j}\|x_i\|^2-2\sum_{i,j}x_i^Tx_j+\sum_{i,j}\|x_j\|^2$. This simplified to $n\sum_i|x_i\|^2-0+n\sum_j\|x_i\|^2=2n\sum_{i}\|x_i\|^2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.