Re: Similar matrices->similar eigenvectors?

*To*: mathgroup at smc.vnet.net*Subject*: [mg80381] Re: Similar matrices->similar eigenvectors?*From*: Ray Koopman <koopman at sfu.ca>*Date*: Mon, 20 Aug 2007 06:05:34 -0400 (EDT)*References*: <fabgkn$ici$1@smc.vnet.net>

u = previous eigenvectors v = current eigenvectors v = (r = u.#; If[r[[Ordering[Abs@r,-1][[1]]]]<0,-#,#])& /@ v will reorient the current eigenvectors to agree with the previous ones. You're on your own the first time thru. Try u = {{1,...,1}}? On Aug 20, 12:42 am, Yaroslav Bulatov <yarosla... at gmail.com> wrote: > In the default implementation of "Eigenvectors", the orientations seem > arbitrary. Changing the matrix slightly could end up flipping the > eigenvectors 180 degrees. A simple fix of telling eigenvectors to > always be on one side of some arbitrary plane doesn't work because it > will flip eigenvectors that are near-parallel to the plane with small > changes in the matrix. > > I'm trying to make a demo of multi-dimensional scaling, and the result > is that as I drag the slider, the points flip back and forth > erratically. > > Basically I'd like to get a function g[mat] which returns eigenvectors > of mat, and is continuous, what is the simplest way of achieving this? > > ----------- > distances2points[d_] := (n = Length[d]; > (*nxn matrix of ones*)j = Table[1, {n, n}]; > (*centering matrix*)h = IdentityMatrix[n] - j/n; > a = -d*d/2; > b = h.a.h; > (*Eigenvectors are returned with arbitrary orientation, > orient them to point in the same halfplane*) > orient[v_, orientvec_] := ((*1,1,1,1 halfplane is often ambiguous, > use random halfplane*)(*SeedRandom[0]; > orientvec=RandomReal[{0,1},Length[v]];*) > If[Round[v, .1].orientvec > 0, -v, v]); > vecs = Eigenvectors[b][[1 ;; 2]]; > (*vecs=orient[#,b[[1]]]&/@vecs;*) > vals = Eigenvalues[b][[1 ;; 2]]; > Point[Re[#]] & /@ Transpose[vecs*Sqrt[vals]]) > Clear[d1, d2, d3, d4, d5, d6]; limits = {{{d1, 0, "1->2"}, 0, > 1}, {{d2, 0, "1->3"}, 0, 1}, {{d3, 0, "1->4"}, 0, > 1}, {{d4, 0, "2->3"}, 0, 1}, {{d5, 0, "2->4"}, 0, > 1}, {{d6, 0, "3->4"}, 0, 1}}; > Manipulate[ > Graphics[distances2points[{{0, d1, d2, d3}, {d1, 0, d4, d5}, {d2, d4, > 0, d6}, {d3, d5, d6, 0}}], PlotRange -> {{-1, 1}, {-1, 1}}], > Evaluate[Sequence @@ limits], LocalizeVariables -> False]