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Groups > comp.lang.forth > #60818
| From | hum9 <obi65537@gmail.com> |
|---|---|
| Newsgroups | comp.lang.forth |
| Subject | Re: fast matrix determinant |
| Date | 2017-06-16 15:58 +0300 |
| Organization | A noiseless patient Spider |
| Message-ID | <oi0ked$l1f$1@dont-email.me> (permalink) |
| References | (6 earlier) <ef709c75-138b-4e9c-ad5b-08d40b24914f@googlegroups.com> <ohu72p$nob$1@dont-email.me> <1435c2f6-0633-4ed3-9dd1-e68d30442114@googlegroups.com> <oi00ml$lab$1@dont-email.me> <cc86be22-56e3-4816-9a13-517ec5f2af9c@googlegroups.com> |
On 16.06.2017 13:38, krishna.myneni@ccreweb.org wrote:
> On Friday, June 16, 2017 at 2:21:21 AM UTC-5, hum9 wrote:
> ...
>> Thank you for support and for inspiration! :)
>>
>
> Glad to be of help.
>
>> I reformulated that theorem so it looks now:
>>
>> If real NxN matrix [A] has N orthonormal eigenvectors,
>> with matrix [C] of orthonormal eigenvectors as columns.
>>
>> In the case of small perturbation of [A], for every
>> unperturbed eigenvector {x} corresponds an perturbed
>> eigenvector {x+dx} collinear with {x}.
>>
>> Changing basis by [C] to a small perturbed version of [A]
>> will result an slight perturbed diagonal matrix with small
>> perturbed eigenvalues on its diagonal.
>>
>
> I think you're not stating the main result, which is the expression for the first-order correction to the eigenvalues. I would state the result in the following way:
>
> Thm: Given a real matrix A, for which the eigenvectors
> are orthogonal, a matrix C containing the eigenvectors
> as columns, and a perturbation matrix dA, the
> first-order corrections to the eigenvalues of the
> perturbed matrix, A' = A + dA, are given by the
> diagonal elements of the similarity transformation,
>
> B = C^T dA C
>
> (end of theorem)
>
> This restatement of the result is missing the geometric
> interpretation you suggest above, though, and that's
> essential to understanding the result, but not to stating
> the result. It's not clear to me that the transformation,
>
> e' = C e
>
> where e is an eigenvector of A, gives a vector e',
> which is collinear with e.
>
> Also, I think stating the result in this manner makes it
> easy to generalize to the case of complex matrices. In
> any case, you should check the literature to see if this
> result is already well-known to mathematicians.
>
> Krishna
>
Inside proof, it was demonstrated that left side of equality 4) is zero:
for every k=1..N
([A] -Lk*[I]) * {dx}k = 0
That means delta-eigenvector {dx}k is also an eigenvector of [A]!
Corresponding to eigenvalue Lk is eigenvector {x} also {dx}k,
so {x}k is parallel to {dx}k.
That means for an eigenvalue Lk of [A], perturbed eigenvector
{x+dx}k is collinear with unperturbed eigenvector of {x}k :)
hum9
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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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