generalized eignvalues/eignvectors

I want to solve the following generalized eignvalues/eignvector problem:

A*w=D*B*w 

Where A is my first square matrix and B my second one, D is the eignvalues and w contains the eignvectors.

I tried to look at https://www.mathworks.com/help/matlab/ref/eig.html#d120e309738 but I did not get the same form I want.

is it just enough to state my problem as the following:

[w,D]=eig(A,B)

or there is another solution. Any suggestion? Thanks very much

[Bruno Luong]

I think your formulation is wrong (or more milly said not "standard"). The eigen value generalized decomposition is

% right eigen vectors

A*V = B*V*D

% left eigen vectors

W'*A = D*W'*B

Where D is diagonal.

Only with these forms that you can arrange with diagonal element of D (eigen values) and colums of V or W (right/left eigen vectors) to satisfy eigen equations.

In the strange form you wrote there is no separation equation between eigen vectors.

[David Goodmanson]

Hello MA,

For the w matrix, one finds column eigenvectors, each with its own eigenvalue, and concatenates them to produce w. Is it the case that for each eigenevector u and its associated eigenvalue lambda, you are solving the equation

A*u = lamda*B*u?

Because if so, the correct resulting equation is

A*w = B*w*D

where D is the diagonal matrix of eigenvalues, and it multiplies on the right. You obtain the generalized eigenvalue form that is solved by Matlab.

[Bruno Luong]

The left eigen vectors

w'A= lamdda*w'*B % marix equation: W'*A = D*W'*B

can also be returned by MATLAB with 3-output syntax of EIG

[~,D,W] = eig(A,B)

w is columns of W, lambda diagonal elements of D.

Note that both left/right can be also obtained by Schur's decomposition with MATLAB QZ, but this is another topic.

Answer 1

[Asad (Mehrzad) Khoddam]

If you multiply both sides by inv(D) we will have:

B*w = inv(D) * A * B

So you can use:

[w, invD] = eig(B, A);
D = inv(InvD);

[David Goodmanson]

Hi Asad,

you meant to say

B*w = inv(D) * A * w

but unfortunately that is not in the specified form for either the left or right generalized eigenvalue problem.

[Asad (Mehrzad) Khoddam]

Yes, you are right

[MA]

Thanks for your answers.