Re: ? what's wrong with eig()
- From: "Cheng Cosine" <acosine@xxxxxxxxxx>
- Date: Tue, 7 Aug 2007 22:10:30 -0400
"Titus" <titus.edelhofer@xxxxxxxxxxxx> wrote in message
news:f8pmer$ejp$1@xxxxxxxxxxxxxxxxxxxxx
"Cheng Cosine" <acosine@xxxxxxxxxx> schrieb im Newsbeitrag
news:46b0558c$0$12228$4c368faf@xxxxxxxxxxxxxxxxx
Hi:some remarks:
A = hilb(3); B = rand(3); B = B'*B;
[V, D] = eig(A,B)
V =
-0.2160 -0.3228 -1.8779
1.1764 0.2763 0.3645
-1.1168 1.1062 1.4785
D =
0.0072 0 0
0 0.2000 0
0 0 1.7418
[Vi, Di] = eig(inv(B)*A)
Vi =
0.7767 0.2724 0.1320
-0.1508 -0.2331 -0.7189
-0.6115 -0.9335 0.6825
Di =
1.7418 0 0
0 0.2000 0
0 0 0.0072
Why those generalized eigenvectors are not the same? The above two
ways should be mathematically equivalent.
Vi'*Vi
ans =
1.0000 0.8176 -0.2065
0.8176 1.0000 -0.4336
-0.2065 -0.4336 1.0000
V'*V
ans =
2.6779 -0.8408 -0.8169
-0.8408 1.4042 2.3424
-0.8169 2.3424 5.8453
The above should be identity matrix, but why it is not?
For example,
[Va, Da] = eig(A)
Va =
-0.1277 0.5474 0.8270
0.7137 -0.5283 0.4599
-0.6887 -0.6490 0.3233
Da =
0.0027 0 0
0 0.1223 0
0 0 1.4083
Va'*Va
ans =
1.0000 0.0000 0.0000
0.0000 1.0000 0.0000
0.0000 0.0000 1.0000
- I'm not sure, if V'*V and Vi'*Vi should result to the identity, since
inv(B)*A is not symmetrical. Va'*Va is the identity, since A is symmetric.
- Interesting, that Vi is not V, but both answers are right (A*V-B*V*D and
inv(B)*A*Vi-Vi*Di are both zero, and for both you can interchange the
Vi<->V and Di<->D. So the eigenvectors need not be unique. If you take a
closer look at Vi, you will see, that it's indeed V just with a scaling
(and permutation):
DD = D(:, 3:-1:1);
VV = Vi(:, 3:-1:1); % permute columns so DD=D
VV./V % constant factors for columns
Because: if x is an eigenvector, p*x (for p~=0) is an eigenvector as well.
Don't know how did you find relation between VV and Vi, but to me, the
following
test told me two things only: 1) eigenvectors obtained from eig(A,B) when B
is
not identiy matrix have norms more than 1, and 2) eigenvectors for eig(A,B)
and eig(A,eye(size(A))) are not simply rescaling and permutation.
A = rand(3); A = A*A', B = hilb(3)
A =
1.2121 1.3337 0.8867
1.3337 1.7988 1.4752
0.8867 1.4752 1.4989
B =
1.0000 0.5000 0.3333
0.5000 0.3333 0.2500
0.3333 0.2500 0.2000
inv(B)*A
ans =
-10.5047 -8.4967 -0.1603
52.8369 31.8151 -18.4839
-44.1051 -18.2315 30.8669
[V D] = eig(A)
V =
-0.5057 -0.7086 0.4920
0.7431 -0.0681 0.6658
-0.4383 0.7023 0.5610
D =
0.0209 0 0
0 0.4614 0
0 0 4.0275
[Vv Dv] = eig(A,B)
Vv =
-1.6207 -1.3887 2.1083
2.0629 3.3426 -13.2880
-1.1045 -0.0147 13.3709
Dv =
0.2010 0 0
0 9.9451 0
0 0 42.0311
norm(V(:,1))
ans =
1.0000
norm(Vv(:,1))
ans =
2.8465
A*Vv(:,1)/norm(Vv(:,1))
ans =
-0.0676
-0.0282
-0.0173
B*Vv(:,1)/norm(Vv(:,1))*Dv(1,1)
ans =
-0.0676
-0.0282
-0.0173
Or maybe you can show me how you find rescaling and permutation
Thanks,
by Cheng Cosine
Aug/07/2k7 NC
.
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