Re: fast matrix determinant

From	hum9 <obi65537@gmail.com>
Newsgroups	comp.lang.forth
Subject	Re: fast matrix determinant
Date	2017-06-16 17:50 +0300
Organization	A noiseless patient Spider
Message-ID	<oi0r0h$cli$1@dont-email.me> (permalink)
References	(8 earlier) <1435c2f6-0633-4ed3-9dd1-e68d30442114@googlegroups.com> <oi00ml$lab$1@dont-email.me> <03614458-497e-4859-b8e0-89787dfbd353@googlegroups.com> <bad4335a-f78b-4458-b684-096d63be2842@googlegroups.com> <0b41c1ae-45e7-4431-a97b-c190d3b59a7e@googlegroups.com>

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On 16.06.2017 15:40, krishna.myneni@ccreweb.org wrote:
> On Friday, June 16, 2017 at 7:12:34 AM UTC-5, krishna...org wrote:
>> On Friday, June 16, 2017 at 5:59:01 AM UTC-5, m...@iae.nl wrote:
>> ...
>>> The theorem would be more useful if you could extend it
>>> and say something about the relationship between small
>>> changes in A and the resulting changes in D and X.
>>>
>>
>> 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.

hum9

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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