Re: Similar matrices->similar eigenvectors?

*To*: mathgroup at smc.vnet.net*Subject*: [mg80519] Re: Similar matrices->similar eigenvectors?*From*: Yaroslav Bulatov <yaroslavvb at gmail.com>*Date*: Fri, 24 Aug 2007 01:58:23 -0400 (EDT)*References*: <14594541.1187598542998.JavaMail.root@m35>

I've tried both approaches, and they seem to remove some of the jittering, but now I realize that the problem is more fundamental -- spectral decomposition of a matrix with degenerate eigenvalues is ill- posed. As a result, Eigenvectors, and SingularValueDecomposition are unstable for inputs in the vicinity of those points. Is there a standard approach for regularizing the problem so that eigenvector finding becomes stable? Yaroslav On Aug 21, 2:15 am, DrMajorBob <drmajor... at bigfoot.com> wrote: > I don't think it's possible to eliminate all discontinuities, but this may > improve things: > > Clear[normalize, eigensystem] > normalize[v : {(0.) ..., x_?Negative, ___}] := -v/Norm[v] > normalize[v_?VectorQ] := v/Norm[v] > normalize[m_?MatrixQ] := normalize /@ m > eigensystem[m_?MatrixQ] := Module[{e, vecs, vals, o}, > {vals, vecs} = Eigensystem[m]; > vecs = normalize@vecs; > o = Reverse@Ordering@vals; > {vals[[o]], vecs[[o]]} > ] > eigensystem[m_?MatrixQ, k_Integer?Positive] := > eigensystem[m][[All, ;; k]] > > distances2points[d_] := (n = Length[d]; > h = IdentityMatrix[n] - Table[1, {n, n}]/n; > b = -h.(d*d/2).h; > {vals, vecs} = eigensystem[b, 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] > > Replace eigensystem with Eigensystem in "distances2points", to get the > original behavior. > > Bobby > > On Mon, 20 Aug 2007 02:38:50 -0500, 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] > > -- > > DrMajor... at bigfoot.com