Similar matrices->similar eigenvectors?

*To*: mathgroup at smc.vnet.net*Subject*: [mg80371] Similar matrices->similar eigenvectors?*From*: Yaroslav Bulatov <yaroslavvb at gmail.com>*Date*: Mon, 20 Aug 2007 03:38:50 -0400 (EDT)

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]