Finding the coordinates of points from distance matrix

If for points p, q, and r you have pq, qr, and rp in your matrix, you have a triangle.

Wherever you have a triangle in your matrix you can compute one of two solutions for that triangle (independent of a euclidean transform of the triangle on the plane). That is, for each triangle you compute, it's mirror image is also a triangle that satisfies the distance constraints on p, q, and r. The fact that there are two solutions even for a triangle leads to the chirality problem: You have to choose the chirality (orientation) of each triangle, and not all choices may lead to a feasible solution to the problem.

Nevertheless, I have some suggestions. If the number entries is small, consider using simulated annealing. You could incorporate chirality into the annealing step. This will be slow for large systems, and it may not converge to a perfect solution, but for some problems it's the best you and do.

The second suggestion will not give you a perfect solution, but it will distribute the error: the method of least squares. In your case the objective function will be the error between the distances in your matrix, and actual distances between your points.

Step 1, arbitrarily assign one point P1 as (0,0).

Step 2, arbitrarily assign one point P2 along the positive x axis. (0, Dp1p2)

Step 3, find a point P3 such that

Dp1p2 ~= Dp1p3+Dp2p3
Dp1p3 ~= Dp1p2+Dp2p3
Dp2p3 ~= Dp1p3+Dp1p2

and set that point in the "positive" y domain (if it meets any of these criteria, the point should be placed on the P1P2 axis).
Use the cosine law to determine the distance:

cos (A) = (Dp1p2^2 + Dp1p3^2 - Dp2p3^2)/(2*Dp1p2* Dp1p3)
P3 = (Dp1p3 * cos (A), Dp1p3 * sin(A))

You have now successfully built an orthonormal space and placed three points in that space.

Step 4: To determine all the other points, repeat step 3, to give you a tentative y coordinate. (Xn, Yn).
Compare the distance {(Xn, Yn), (X3, Y3)} to Dp3pn in your matrix. If it is identical, you have successfully identified the coordinate for point n. Otherwise, the point n is at (Xn, -Yn).

Note there is an alternative to step 4, but it is too much math for a Saturday afternoon

The answers based on angles are cumbersome to implement and can't be easily generalized to data in higher dimensions. A better approach is that mentioned in my and WimC's answers here: given the distance matrix D(i, j), define

M(i, j) = 0.5*(D(1, j)^2 + D(i, 1)^2 - D(i, j)^2)

which should be a positive semi-definite matrix with rank equal to the minimal Euclidean dimension k in which the points can be embedded. The coordinates of the points can then be obtained from the k eigenvectors v(i) of M corresponding to non-zero eigenvalues q(i): place the vectors sqrt(q(i))*v(i) as columns in an n x k matrix X; then each row of X is a point. In other words, sqrt(q(i))*v(i) gives the ith component of all of the points.

The eigenvalues and eigenvectors of a matrix can be obtained easily in most programming languages (e.g., using GSL in C/C++, using the built-in function eig in Matlab, using Numpy in Python, etc.)

Note that this particular method always places the first point at the origin, but any rotation, reflection, or translation of the points will also satisfy the original distance matrix.

This is a math problem. To derive coordinate matrix X only given by its distance matrix.

However there is an efficient solution to this -- Multidimensional Scaling, that do some linear algebra. Simply put, it requires a pairwise Euclidean distance matrix D, and the output is the estimated coordinate Y (perhaps rotated), which is a proximation to X. For programming reason, just use SciKit.manifold.MDS in Python.