Generalized eigenvalue problem

Question

I'm trying to convert a generalized eigenvalue problem into a normal eigenvalue calculation.

I have this code:

[V,D,flag] = eigs(A, T);

Now I convert it into:

A1 = inv(T)*A;
[V1,D1,flag1] = eigs(A1);

Shouldn't I get the same result? From what I understand in the Matlab documentation, the first equation solves:

A*V = B*V*D

and the second one solves:

A*V = V*D

am I missing something?

Thanks!!

Answer 1

A quick example:

A = rand(4); B = randn(4); B = B'*B;         %'# some matrices
[VV,DD] = eig(B\A);
[V,D] = eigs(A,B);
V = bsxfun(@rdivide, V, sqrt(sum(V.*V)));    %# make: norm(V(:,i))==1

The result:

V =
     -0.64581       0.8378      0.77771      0.50851
      0.70571     -0.51601     -0.32503     -0.70623
      0.27278     0.076874     -0.51777      0.25359
      0.10245      0.16095     -0.14641     -0.42232
VV =
     -0.64581       0.8378     -0.77771     -0.50851
      0.70571     -0.51601      0.32503      0.70623
      0.27278     0.076874      0.51777     -0.25359
      0.10245      0.16095      0.14641      0.42232
D =
       17.088            0            0            0
            0      0.27955            0            0
            0            0     -0.16734            0
            0            0            0     0.027889
DD =
       17.088            0            0            0
            0      0.27955            0            0
            0            0     -0.16734            0
            0            0            0     0.027889

Note: The eigenvalues are not always sorted the same, also the sign convention might be different...