[Scilab-users] eigs calculation

[Carrico, Paul]

Dear Tim,

Thanks for the enlightenment, but unfortunately I confess I don't know how to proceed now; 

I feel that I need to fix such issue; another example if needed (see screenshot in attachment), the product herebellow should lead to the Identity matrix

Verif1 = u' * M * u 

(vhere u is the eigenvector matrix)



Paul


-----Message d'origine-----
De : users [mailto:users-bounces at lists.scilab.org] De la part de Tim Wescott Envoyé : vendredi 19 juin 2015 18:38 À : International users mailing list for Scilab.
Objet : Re: [Scilab-users] eigs calculation

"Normalization" refers to jiggering the numbers around in a way that does not change the problem in a strictly mathematical sense, but which makes it more tractable.

d = eigs(A, B) computes the solutions to

A * v = lambda * B * v

So any nonsingular square matrix N won't change the problem if it's multiplied in:

N * A * v = N * lambda * B * v

Because lambda is a scalar (well, I hope I'm getting that right) you can change the problem to

A_ = N * A, B_ = N * B, and

d = eigs(A_, B_)

will, theoretically, get the same answers as with the original matrices, but possibly with better numerical conditioning.  This is what you're doing when you change units.

If you don't mind the meaning of your eigenvectors changing, you can do a similarity transform.  Start with

N * A * N^(-1) * N * v = N * lambda * B * N^(-1) * N * v

Now set

A_ = N * A * N^(-1)
B_ = N * B * N^(-1)
v_ = N * v

[d, v_] = eigs(A_, B_); v = N^(-1) * v_;

will, again, theoretically give you the same numbers as before, but it may be better conditioned numerically.

Actually _finding_ N, or giving you advise on how to do so, is beyond my powers -- but maybe this will set you on a better road.

-- 

Tim Wescott
www.wescottdesign.com
Control & Communications systems, circuit & software design.
Phone: 503.631.7815
Cell:  503.349.8432


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