[Scilab-users] eigs calculation

[tim at wescottdesign.com]

That's not at all atypical, and it's what you expect from a numerical 
process to arrive at an eigenvector matrix.  In the context of ones on 
the main diagonal, 1e-16 is "mostly zero".

On 2015-06-22 03:08, Carrico, Paul wrote:
> 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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