Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > comp.lang.forth > #60886

Re: fast matrix determinant

Show all headers | View raw

On 16.06.2017 23:39, krishna.myneni@ccreweb.org wrote:
> On Friday, June 16, 2017 at 9:50:21 AM UTC-5, hum9 wrote:
>> On 16.06.2017 15:40, krishna...org wrote:
> ...
>>>> Hum9 omitted the main result in his reformulation statement, namely, the diagonal elements of B
>>>>
>>>>     B = C^T dA C
>>>>
>>>> provide the quantitative relationship between small changes in A and the changes in D, to first order. This appears to be rigorous (and provable) when the eigenvectors of A are orthogonal.
>>>
>>>
>>> A clearer way to see this, and which also makes it
>>> obvious, is to write
>>>
>>>       D  =  C^T A C
>>>
>>> where C is the matrix of eigenvectors.
>>>
>>>       D + dD ≈ C^T A + dA C
>>>
>>> Thus,
>>>
>>>       dD  ≈  C^T dA C
>>>
>>> where only the diagonal elements are retained for the resulting matrix on the R.H.S.
>>>
>>> In this form, the result can obviously be extended to real matrices which do not have orthogonal eigenvectors, by writing,
>>>
>>>        dD ≈ C^-1 dA C
>>>
>>> Krishna
>>>
>>>
>>>
>>
>> In mentioned context, yes, that is a *quantitative* one,
>> plus another *qualitative* consequence:
>> for every eigenvector of unperturbed matrix [A],
>> difference between unperturbed eigenvector {x} and
>> perturbed eigenvector {x+dx} is only by magnitude,
>> not by orientation.
>>
> 
> Your "qualitative" result may be true in first-order, for the specific matrix case considered, but it will not be true for general perturbations for 2nd order and higher order. I guess I don't understand what you consider to be the essential result. I had interpreted the essential result to be the first-order approximation to the eigenvalues of the perturbed matrix. This is a result from perturbation theory, which goes back a long way. The wikipedia page on eigenvalue perturbation theory has some issues, but you can find your result on that page:
> 
> https://en.wikipedia.org/wiki/Eigenvalue_perturbation
> 
> The general theory provides a means of calculating not only the first order eigenvalues of the perturbed matrix, but also the first order eigenvectors of the perturbed matrix. An iterative procedure can be used to compute higher order corrections. If I find a decent reference online to point you to, I'll post it.
> 
> Regards,
> Krishna
> 
> 
> 

I reviewed proof paper and found that equation
{x}k^T * ([A] -Lk*[I]) * {dx}k = 0
   I copied wrongly from paper to computer, as
([A] -Lk*[I]) * {dx}k = 0

As a "GIGO theorem" applies, collinearity result is G(arbage).
*My apologies*.


Yet is something interesting in solving for delta eigenvector {dx}k,

([A] -Lk*[I]) * {dx}k = ([I]*dLk - [dA]) * {x}k

as Determinant([A] -Lk*[I]) = 0.

I note [B] = [C]^T * [dA] * [C] and
transformed {dx}k into orthonormal basis -> {dy}k = [C]^T*{dx}k
with [C]^-1=[C]^T.

Transforming equation in orthonormal basis of eigenvectors, give
the solution:

dYik = |{x}k| * Bik/(Lk - Li) , for Li <> Lk

undetermined for rest, 0 * dYkk = 0.

That means components perpendicular to {x}k could be calculated,
but along the {x}k could be every finite number.

I checked Wikipedia, I found that is presented a more general case,
but in general they took the same approach as I, without factoring
terms, which I did.

Thank you for kind support,
hum9

Back to comp.lang.forth | Previous | Next — Previous in thread | Next in thread | Find similar | Unroll thread

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

csiph-web