Suppose a matrix $D\in\mathbb{R}^{n\times n}$ of Euclidean distances between $n$ points is given. To obtain a Gram matrix (matrix of inner-products of points that give rise to distances in $D$), one performs $$G=-1/2\cdot PD^{(2)}P^T,$$ where $D^{(2)}$ contains squared entries of $D$, and $P=(I_n-1_nw^T)$, $w^T1_n=1$. It is known (see http://www.sciencedirect.com/science/article/pii/0024379585901879) that in case $w^T1_n=1$, the rank of all Gram matrices is identical, and corresponds to true dimensionality of the data.However, I wonder what happens when $w^T1_n\neq 1$. Could there be a choice of two different $P$ for which the ranks of associated $G$ are different?also, why is the constraint $w^T1_n=1$ imposed?
Configuration shift to change the rank of a Gram matrix
linear algebra;matrices;high dimensional