       Eigensystem calculated for Integer versus Real numbers.

• To: mathgroup at smc.vnet.net
• Subject: [mg5383] Eigensystem calculated for Integer versus Real numbers.
• From: Wouter Meeussen <w.meeussen.vdmcc at vandemoortele.be>
• Date: Thu, 5 Dec 1996 14:50:22 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```hi all,

who can help me understand the difference between the Eigensystem as
calculated for Integer versus Real numbers :

define any integer symmetric matrix :

in   m=(m=Table[Random[Integer,{0,1}],{3},{3}])+Transpose[m]
out         {{2, 2, 0}, {2, 0, 2}, {0, 2, 2}}

be shure to pick one with non-zero determinant:

in   Det[m]
out         -16

in   {val,vec}=Eigensystem[m]
out         {{-2, 2, 4}, {{1, -2, 1}, {-1, 0, 1}, {1, 1, 1}}}

now, feed Eigensystem with reals:

in   {nval,nvec}=Eigensystem[m//N]//Chop
out         {{4., -2., 2.},

the ordering can be set 'right' by:
(left as an exercise to ...)

in   {nval,nvec}=Transpose at Sort@Transpose at {nval,nvec}//Chop
out         {{-2., 2., 4.},

at this point,both sets of eigenvectors are different by the factors :

(* cleanup . nvec == vec *)

in   cleanup=vec.Inverse[nvec]//Chop
out         {{2.44949, 0, 0},
{0, 1.41421, 0},
{0, 0, 1.73205}}

I could understand this in case of degeneracy (not all eigenvalues different),
but here ? Have I overlooked something ?

Check it out with this program, if you want :

program:=Module[{},m=(k=Table[Random[Integer,{0,4}],{3},{3}])+Transpose[k];

Wouter.

(  baffled, as usual		)
(  and this is not a signature!	)
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```

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