Re: fast matrix determinant

Newsgroups	comp.lang.forth
Date	2017-06-14 11:05 -0700
References	<0b7ec571-fab4-452c-97e5-e2b629d79c38@googlegroups.com> <1b7c433f-3273-4942-b620-958e31e19b0f@googlegroups.com> <adf343c7-467e-4e6c-b4e1-5d72f3d9ece3@googlegroups.com> <b6974c55-fda0-4102-8630-71522e32e842@googlegroups.com> <ohrj4a$jra$1@dont-email.me>
Message-ID	<d8249bbf-eccb-4fcd-9391-df75d571cab9@googlegroups.com> (permalink)
Subject	Re: fast matrix determinant
From	krishna.myneni@ccreweb.org

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On Wednesday, June 14, 2017 at 10:05:08 AM UTC-5, hum9 wrote:
> On 12.06.2017 17:14, krishna...org wrote:
...
> Recently, after your post, I have found something that
> could be useful:
> 
> 
> A symbol between '[' ']' represents a matrix,
> between '{' '}' represents vector,
> ^T   means transpose
> ^-1  means inverse of matrix.
> 
> Theorem (proposition):
> 
> Be a real symmetric matrix [A] NxN  with N distinct nonzero eigenvalues
> and [C] its transformation matrix into orthonormal basis of [A]
> eigenvectors.
> 
> Then, for very small perturbation [dA] (Norm([dA] << Norm([A]))
> will result delta eigenvalues dL, for every index k, 1 <= k <= N
> 
> dLk = Bkk  ,where [B] = [C]^T * [dA] * [C]
> 
> ...

That's an interesting result, but I'm not sure how it can be applied to the problem I described. I don't understand what your transformation matrix, [C], is supposed to do? Does it transform basis vectors into a new basis set? If so, the transformation,

   [B] = [C]^T * [dA] * [C]

is like a similarity transformation, for the special case of

   [C]^-1 = [C]^T

I haven't examined your proof in detail, but if one can obtain a first order approximation to the new eigenvalues, based on perturbations (small changes) to the elements of [A], it seems useful.

In my problem description, the matrix [H] ([A] in your description) is not being perturbed, i.e. small changes are not being made to the matrix elements, but it's rank is being increased by 1, in successive approximations.

Krishna

Thread

fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-10 10:29 -0700
  Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-12 05:41 -0700
    Re: fast matrix determinant mhx@iae.nl - 2017-06-12 06:03 -0700
      Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-12 07:14 -0700
        Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-12 07:28 -0700
        Re: fast matrix determinant mhx@iae.nl - 2017-06-12 11:21 -0700
          Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-12 16:39 -0700
        Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-14 18:05 +0300
          Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-14 11:05 -0700
            Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-14 21:37 +0300
          Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-15 09:59 +0300
            Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-15 03:30 -0700
              Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-15 17:57 +0300
                Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-15 19:06 -0700
                Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-16 10:21 +0300
                Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-16 03:38 -0700
                Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-16 15:58 +0300
                Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-16 06:04 -0700
                Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-16 16:19 +0300
                Re: fast matrix determinant mhx@iae.nl - 2017-06-16 03:58 -0700
                Re: fast matrix determinant albert@cherry.spenarnc.xs4all.nl (Albert van der Horst) - 2017-06-16 13:39 +0200
                Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-16 05:58 -0700
                Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-16 05:12 -0700
                Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-16 05:40 -0700
                Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-16 17:50 +0300
                Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-16 13:39 -0700
                Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-17 10:01 +0300
                Re: fast matrix determinant mhx@iae.nl - 2017-06-17 01:31 -0700
                Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-17 15:24 -0700
            Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-15 17:59 +0300
          Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-16 10:11 +0300
            Re: fast matrix determinant hum9 <obi65537@gmail.com> - 2017-06-17 09:20 +0300
      Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-13 16:43 -0700
        Re: fast matrix determinant mhx@iae.nl - 2017-06-13 22:04 -0700
        Re: fast matrix determinant mhx@iae.nl - 2017-06-14 12:23 -0700
          Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-14 18:31 -0700
    Re: fast matrix determinant krishna.myneni@ccreweb.org - 2017-06-15 19:13 -0700

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