Eigenshuffle

Consistently sorted eigenvalue and eigenvector sequences

A problem that I've seen many times on the newsgroup is how eig returns its eigenvalues and eigenvectors. By itself, eig returns an arbitrary order for the eigenvalues and eigenvectors. They are often nearly sorted in order, but this is not assured. The other issue is the eigenvectors can have an arbitrary sign applied to them.

Worse, when you have a sequence of eigenvalue problems, the eigenvalues can sometimes cross over. One would like to sort the eigenvalues/eigenvectors so the sequence is consistent.

I've designed eigenshuffle.m to do exactly that. It takes a pxpxn array, where each page of the array is one matrix where we wish to compute the eigenvalues. Eigenshuffle tries to permute the eigenvalues and eigenvectors to be maximally consistent from one step in the sequence to the next. Eigenshuffle also chooses the sign to be applied to each eigenvector to be maximally consistent with the the vectors prior to it in the sequence of eigenproblems.

As an example, try this simple matrix function of a parameter t.

Efun = @(t) [1 2*t+1 t^2 t^3;2*t+1 2-t t^2 1-t^3; ...
t^2 t^2 3-2*t t^2;t^3 1-t^3 t^2 4-3*t];

Aseq = zeros(4,4,21);
for i = 1:21
Aseq(:,:,i) = Efun((i-11)/10);
end
[Vseq,Dseq] = eigenshuffle(Aseq);

To see that eigenshuffle has done its work correctly,
look at the eigenvalues in sequence after the shuffle.

t = (-1:.1:1)';
[t,Dseq']
ans =
-1 8.4535 5 2.3447 0.20181
-0.9 7.8121 4.7687 2.3728 0.44644
-0.8 7.2481 4.56 2.3413 0.65054
-0.7 6.7524 4.3648 2.2709 0.8118
-0.6 6.3156 4.1751 2.1857 0.92364
-0.5 5.9283 3.9855 2.1118 0.97445
-0.4 5.5816 3.7931 2.0727 0.95254
-0.3 5.2676 3.5976 2.0768 0.858
-0.2 4.9791 3.3995 2.1156 0.70581
-0.1 4.7109 3.2 2.1742 0.51494
0 4.4605 3 2.2391 0.30037
0.1 4.2302 2.8 2.2971 0.072689
0.2 4.0303 2.5997 2.3303 -0.16034
0.3 3.8817 2.4047 2.3064 -0.39272
0.4 3.8108 2.1464 2.2628 -0.62001
0.5 3.8302 1.8986 2.1111 -0.83992
0.6 3.9301 1.5937 1.9298 -1.0537
0.7 4.0927 1.2308 1.745 -1.2685
0.8 4.3042 0.82515 1.5729 -1.5023
0.9 4.5572 0.40389 1.4272 -1.7883
1 4.8482 -8.0012e-16 1.3273 -2.1755

Here, columns 2:5 are the shuffled eigenvalues. See that the second eigenvalue goes to zero, but the third eigenvalue remains positive. We can plot eigenvalues and see that they have crossed, near t = 0.35 in Efun.

plot(-1:.1:1,Dseq')

For a better appreciation of what eigenshuffle did, compare the result of eig directly on Efun(.3) and Efun(.4). Thus:

[V3,D3] = eig(Efun(.3))
V3 =
-0.74139 0.53464 -0.23551 0.3302
0.64781 0.4706 -0.16256 0.57659
0.0086542 -0.44236 -0.89119 0.10006
-0.17496 -0.54498 0.35197 0.74061

D3 =
-0.39272 0 0 0
0 2.3064 0 0
0 0 2.4047 0
0 0 0 3.8817

[V4,D4] = eig(Efun(.4))
V4 =
-0.73026 0.19752 0.49743 0.42459
0.66202 0.21373 0.35297 0.62567
0.013412 -0.95225 0.25513 0.16717
-0.16815 -0.092308 -0.75026 0.63271

D4 =
-0.62001 0 0 0
0 2.1464 0 0
0 0 2.2628 0
0 0 0 3.8108

With no sort or shuffle applied, look at V3(:,3). See that it is really closest to V4(:,2), but with a sign flip. Since the signs on the eigenvectors are arbitrary, the sign is changed, and the most consistent sequence will be chosen. By way of comparison, see how the eigenvectors in Vseq have been shuffled, the signs swapped appropriately.

Vseq(:,:,14)
ans =
0.3302 0.23551 -0.53464 0.74139
0.57659 0.16256 -0.4706 -0.64781
0.10006 0.89119 0.44236 -0.0086542
0.74061 -0.35197 0.54498 0.17496

Vseq(:,:,15)
ans =
0.42459 -0.19752 -0.49743 0.73026
0.62567 -0.21373 -0.35297 -0.66202
0.16717 0.95225 -0.25513 -0.013412
0.63271 0.092308 0.75026 0.16815

Note that sequences of generalized eigenvalue problems can now be solved too.

With many thanks to Yi Cao, I've included munkres by permission as a subfunction here.

http://www.mathworks.com/matlabcentral/fileexchange/20652