Convert distance matrix to Vector matrix. The question is following.
In d-dimensional space, there are n-points.
Suppose that we know distance matrix.
Since rotation and reflection do not change distance, fix the
origin of the coordinate system so that the centroid of the set of points is at the origin.
I already find out that how to representing each dot product of two points by elements of distance matrix.
So I can make Gram matrix from distance matrix.
However, I'm stuck with algorithm for determine n- given vectors.
Is there any hints? Thanks in advance.
 A: Not an answer, but too long for a comment.
i. Consider the points in the plane $(0, 1)$, $(0, -1)$. Their distance matrix looks like
$$
\pmatrix{0 & 2 \\ 2 & 0}
$$
and their centroid is the origin.
ii. Consider the points in the plane $(1, 0)$, $(-1, 0)$. Their distance matrix looks like
$$
\pmatrix{0 & 2 \\ 2 & 0}
$$
and their centroid is the origin.
Now given the distance matrix, and all the points "centered at the origin," which they already are, which of the two original sets of points should be the output? The answer: either one. In fact, if you have any set of points (represented as vectors from the origin) $v_1, \ldots, v_n$ and a matrix $M \in O(d)$, the vectors $Mv_1, \ldots, Mv_n$ will produce the same distance matrix.
So "normalizing" by putting the centroid at the origin wiped out $d$ degrees of ambiguity in the answer, but ignored the other $\frac{d(d-1)}{2}$ degrees of freedom. You probably want to think of some way to fix those before asking for "the" vectors that determine this distance matrix.
