Eigenvectors of numerical matrix

Question

I have a large numerical matrix whose eigenvalues are all distinct. In the documentation for Eigenvectors it says:

For approximate numerical matrices m, the eigenvectors are normalized.

Eigenvectors with numeric eigenvalues are sorted in order of decreasing absolute value of their eigenvalues.

So from what I understand, Eigenvectors should give the same result every time. But in reality it gives different results, if I did some heavy calculation tasks in between to evaluations so that the autosaved evaluation results are erased. Why is that?

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Here is the matrix causing the problem. Simply Import it into Mathematica and apply Eigenvectors. I'm using "10.0 for Microsoft Windows (32-bit) (June 30, 2014)".

omega = Import["Omega.m"]
Eigenvectors[omega]

Note that there must be lots of irrelevant calculation in between two evaluations in order to reproduce the problem. Sorry that I can't provide the code that I'm using but I think any heavy calculation will do. Or is there some way to turn off the automatic memoization?

Answer 1

Here is a small trick. Let say you don't know the ordering of your eigenvetors.

m = RandomReal[1, {10, 10}];
m = m + Transpose[m];
es = Eigensystem[m];
es[[1]]

{9.46387, -2.07941, 1.87017, -1.69204, 1.6247, -1.00992, 0.835742, \ -0.786536, 0.669689, -0.463195}

Definitely not ordered. So you can force the order by Sort

es1 = Transpose[Sort[Transpose[es]]];
es1[[1]]

{-2.07941, -1.69204, -1.00992, -0.786536, -0.463195, 0.669689, \ 0.835742, 1.6247, 1.87017, 9.46387}

And it will assure the order of the eigenvectors as well.

Now if you get the feeling that the eigenvectors are not normalised, you can force that as well

es1 = Transpose[Sort[Transpose[es]/. {x_, y_List} :> {x, Normalize[y]}]];

Answer 2

As per comment by @bill, I have not been able to reproduce the problem. I did a trial run using the original matrix using the code below, and the figure values seem fixed. However there are a no. of eigenvectors with low probability of existence (see first figure), and might give rise to problems.

ListPlot[Eigenvectors[N[mat]], Frame -> True, 
 FrameLabel -> {Style["No.", Large, Bold], 
       Style["Occupation Amplitudes", Large, Bold], 
   Style["", Large, Bold]}, Ticks -> Automatic, 
   LabelStyle -> Directive[Black, Bold, Large]]

ListPlot[Eigenvalues[N[mat]], Frame -> True, Ticks -> Automatic, 
 LabelStyle -> Directive[Black, Bold, Large]]

Answer 3

To save your results, linking eigenvectors to the right ordered eigenvalues use:

omega = Import["Omega.m"]
{evals,evecs}:=Eigensystem[N[omega]]

Now the eigenvalues are associated with theire corresponding eigenvectors. After that I suggest to make sure youre eigenvectors are orthogonal. You can do a quick check by:

 MatrixForm[Chop[Table[evecs[[i]].evecs[[j]], {i, 1, Length[evecs]}, {j, 1,Length[evecs]}]]