Re: How to plot a graph, using a distance matrix

*To*: mathgroup at smc.vnet.net*Subject*: [mg89035] Re: How to plot a graph, using a distance matrix*From*: Ray Koopman <koopman at sfu.ca>*Date*: Sat, 24 May 2008 03:54:01 -0400 (EDT)*References*: <g15qni$pdj$1@smc.vnet.net>

On May 23, 12:11 am, Eric Heiman <ehei... at mailinator.com> wrote: > My dilemna is as such: > > I have a matrix (it happens to be 21x21, but don't worry about that) which contains distances between points. > So column 1 has distances from point 1 to every other point, with row 1 being 0 (because the distance to itself is zero). > > What I am wondering is how I would be able to get mathematica to plot a graph of these points. > > Thanks in advance! First the algebra: Let C = (-1/2)(I - uu'/n)B(I - uu'/n), where B is the matrix of squared distances among the n points, I is the identity matrix, u is a column vector whose elements are all 1s, and ' denotes transposition. If the points lie in an m-dimensional space then C will be positive semidefinite, with rank m. Let F be any factoring of such a C, so that FF' = C. Then F contains the coordinates of the points. This is a well-known result in multidimensional scaling. In Mathematica code, if dis is the matrix of distances, let {vals,vex} = Eigensystem[-.5 (#-Mean@#&) /@ Transpose[#-Mean@#& /@ (dis^2)]]; The first m elements of vals should be positive; the rest should be approximate zeros. Then f = Transpose[Sqrt[Take[vals,m]]*Take[vex,m]] will contain the coordinates of the points. However, this assumes that the eigenvectors are orthonormal, which is usually true, but I have not been able to find it in the documentation, so caveat user.