Re: Re: Similar matrices->similar eigenvectors?

*To*: mathgroup at smc.vnet.net*Subject*: [mg80537] Re: [mg80519] Re: Similar matrices->similar eigenvectors?*From*: DrMajorBob <drmajorbob at bigfoot.com>*Date*: Sun, 26 Aug 2007 02:50:45 -0400 (EDT)*References*: <14594541.1187598542998.JavaMail.root@m35> <29244674.1187940243127.JavaMail.root@m35>*Reply-to*: drmajorbob at bigfoot.com

If there were a standard way to make Eigensystem stable in that sense, I suspect WRI would be using it already. What those guys don't know about algebra is rarely worth knowing. Bobby On Fri, 24 Aug 2007 00:58:23 -0500, Yaroslav Bulatov <yaroslavvb at gmail.com> wrote: > 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 > > > > -- DrMajorBob at bigfoot.com