Untitled

Let's start with a known set of eigenvalues and an orthogonal eigenvector basis.

D = diag([-6 -1 2 4 5]);
[V,R] = qr(randn(5));
A = V*D*V';    % note that V' = inv(V)

Now we will take the QR factorization and just reverse the factors.

[Q,R] = qr(A);
A = R*Q;

It turns out that this is a similarity transformation, so the eigenvalues are unchanged.

eig(A)
ans = 
  -6.000000000000001
  -1.000000000000002
   5.000000000000003
   4.000000000000003
   2.000000000000000

What's remarkable is that if we repeat the transformation many times, the process converges to

for k = 1:15
    [Q,R] = qr(A);
    A = R*Q;
end
A
A = 
  -5.717280803318202   1.740123893057575  -0.037768561192074  -0.000000282468282  -0.000000000002784
   1.740123893057576   4.713463869317748  -0.056979263094775  -0.000001252743229   0.000000000030252
  -0.037768561192070  -0.056979263094776   4.003816933941975   0.000010798262392   0.000000000433793
  -0.000000282468282  -0.000001252743229   0.000010798262392   1.999999997528164  -0.000087126218965
  -0.000000000002784   0.000000000030252   0.000000000433792  -0.000087126218965  -0.999999997469674