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Re: Intersection over an index

• To: mathgroup at smc.vnet.net
• Subject: [mg128415] Re: Intersection over an index
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Thu, 18 Oct 2012 02:34:26 -0400 (EDT)
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• References: <20121017001211.0C48B685E@smc.vnet.net>

intersectM[m1_?MatrixQ, m2_?MatrixQ] :=
  Select[m2, MemberQ[m1, #] &];

intersectEV[m : {__?MatrixQ}] := Module[
  {ev = Eigenvectors /@ m},
  Fold[intersectM[#1, #2] &, First[ev], Rest[ev]]]

A[1] = {{-1, -3, 1}, {0, -3, 0}, {-1, -1, -1}};

Eigenvectors[A[1]]

{{1, 1, 1}, {-I, 0, 1}, {I, 0, 1}}

A[2] = {{-2, -1, 1}, {0, -1, 0}, {-1, 1, -2}};

Eigenvectors[A[2]]

{{-I, 0, 1}, {I, 0, 1}, {0, 1, 1}}

A[3] = {{-2, -1, -1}, {0, -1, 0}, {1, -1, -2}};

Eigenvectors[A[3]]

{{I, 0, 1}, {-I, 0, 1}, {0, -1, 1}}

A[4] = {{-2, -1, 1}, {0, -1, 0}, {1, -1, -2}};

Eigenvectors[A[4]]

{{-1, 0, 1}, {1, 0, 1}, {0, 0, 0}}

The first three have common eigenvectors

intersectEV[Table[A[k], {k, 3}]]

{{I, 0, 1}, {-I, 0, 1}}

Adding the fourth does not

intersectEV[Table[A[k], {k, 4}]]

{}


Bob Hanlon


On Tue, Oct 16, 2012 at 8:12 PM, Geoffrey Eisenbarth
<geoffrey.eisenbarth at gmail.com> wrote:
> Given a set of n many matrices A[k], I'd like to find any common eigenvectors. Using
>
> Intersection[Table[Eigenvalues[A[k]],{k,1,n}] doesn't seem to work. For instance:
>
> A[1] = {{-1, -3, 1}, {0, -3, 0}, {-1, -1, -1}};
> A[2] = {{-2, -1, 1}, {0, -1, 0}, {-1, 1, -2}};
> Intersection[Table[A[p], {p, 1, 2}]]
>
> gives me
> {{{-2, -1, 1}, {0, -1, 0}, {-1, 1, -2}}, {{-1, -3, 1}, {0, -3,
>    0}, {-1, -1, -1}}}
>
>
> Any suggestions?
>

• References:
  • Intersection over an index
    • From: Geoffrey Eisenbarth <geoffrey.eisenbarth@gmail.com>