[Numpy-discussion] Getting non-normalized eigenvectors from generalized eigenvalue solution?
Fahreddın Basegmez
mangabasi at gmail.com
Tue Dec 20 23:14:48 EST 2011
Sorry about that. I don't think that terminology is commonly used. This
is what I mean.
Let's say I solve the equations and compute the eigenvalues and
eigenvectors for the given two matrices. I call these results
"non-normalized". Then they can be normalized. Once they are normalized if
I multiply them by any scalar they would become "un-normalized". They
would still be eigenvectors but not necessarily the "non-normalized" ones.
I think there is a specific algorithm that Matlab uses to solve them but I
do not know what that algorithm is.
On Tue, Dec 20, 2011 at 11:01 PM, Olivier Delalleau <shish at keba.be> wrote:
> Ok well I'm sorry, I have no idea what would be the difference between
> "non-normalized" and "un-normalized".
>
> In PCA you may decide to scale your eigenvectors by the inverse of the
> square root of their corresponding eigenvalue so that your projected data
> has unit variance, but it doesn't seem to be what you're after.
>
> Can you point to a link that explains what are the "non-normalized
> eigenvectors" in your application?
>
>
> -=- Olivier
>
> 2011/12/20 Fahreddın Basegmez <mangabasi at gmail.com>
>
>> I think I am interested in the non-normalized eigenvectors not the
>> un-normalized ones. Once the eig function computes the generalized
>> eigenvectors I would like to use them as they are.
>> I would think this would be a common request since the normal-mode
>> frequency response is used in many different fields like chemical and
>> biomolecular sciences as well as engineering and physics. Mathematically
>> there may be no difference between the normalized and non-normalized
>> eigenvectors but physically there is. In my case those values represent
>> deflections. Advantage of the normal-modes is you can apply damping in
>> each direction independent of each other. Amount of damping we apply may
>> be dependent on those deflections so I would need to use the non-normalized
>> results.
>>
>>
>> On Tue, Dec 20, 2011 at 10:15 PM, Olivier Delalleau <shish at keba.be>wrote:
>>
>>> What I don't get is that "un-normalized" eigenvectors can be pretty much
>>> anything. If you care about the specific output of Matlab / Octave, it
>>> means you understand the particular "un-normalization" that these programs
>>> use. In that case you should be able to recover it from the normalized
>>> output from numpy.
>>>
>>>
>>> -=- Olivier
>>>
>>> 2011/12/20 Fahreddın Basegmez <mangabasi at gmail.com>
>>>
>>>> I don't think I can do that. I can go to the normalized results but
>>>> not the other way.
>>>>
>>>>
>>>> On Tue, Dec 20, 2011 at 9:45 PM, Olivier Delalleau <shish at keba.be>wrote:
>>>>
>>>>> Hmm, sorry, I don't see any obvious logic that would explain how
>>>>> Octave obtains this result, although of course there is probably some
>>>>> logic...
>>>>>
>>>>> Anyway, since you seem to know what you want, can't you obtain the
>>>>> same result by doing whatever un-normalizing operation you are after?
>>>>>
>>>>>
>>>>> -=- Olivier
>>>>>
>>>>> 2011/12/20 Fahreddın Basegmez <mangabasi at gmail.com>
>>>>>
>>>>>> I should include the scipy response too I guess.
>>>>>>
>>>>>>
>>>>>> scipy.linalg.eig(STIFM, MASSM)
>>>>>> (array([ 3937.15984097+0.j, 3937.15984097+0.j, 3937.15984097+0.j,
>>>>>> 3923.07692308+0.j, 3923.07692308+0.j, 7846.15384615+0.j]),
>>>>>> array([[ 1., 0., 0., 0., 0., 0.],
>>>>>> [ 0., 1., 0., 0., 0., 0.],
>>>>>> [ 0., 0., 1., 0., 0., 0.],
>>>>>> [ 0., 0., 0., 1., 0., 0.],
>>>>>> [ 0., 0., 0., 0., 1., 0.],
>>>>>> [ 0., 0., 0., 0., 0., 1.]]))
>>>>>>
>>>>>> On Tue, Dec 20, 2011 at 9:14 PM, Fahreddın Basegmez <
>>>>>> mangabasi at gmail.com> wrote:
>>>>>>
>>>>>>> If I can get the same response as Matlab I would be all set.
>>>>>>>
>>>>>>>
>>>>>>> Octave results
>>>>>>>
>>>>>>> >> STIFM
>>>>>>> STIFM =
>>>>>>>
>>>>>>> Diagonal Matrix
>>>>>>>
>>>>>>> 1020 0 0 0 0 0
>>>>>>> 0 1020 0 0 0 0
>>>>>>> 0 0 1020 0 0 0
>>>>>>> 0 0 0 102000 0 0
>>>>>>> 0 0 0 0 102000 0
>>>>>>> 0 0 0 0 0 204000
>>>>>>>
>>>>>>> >> MASSM
>>>>>>> MASSM =
>>>>>>>
>>>>>>> Diagonal Matrix
>>>>>>>
>>>>>>> 0.25907 0 0 0 0 0
>>>>>>> 0 0.25907 0 0 0 0
>>>>>>> 0 0 0.25907 0 0 0
>>>>>>> 0 0 0 26.00000 0 0
>>>>>>> 0 0 0 0 26.00000 0
>>>>>>> 0 0 0 0 0 26.00000
>>>>>>>
>>>>>>> >> [a, b] = eig(STIFM, MASSM)
>>>>>>> a =
>>>>>>>
>>>>>>> 0.00000 0.00000 0.00000 1.96468 0.00000 0.00000
>>>>>>> 0.00000 0.00000 0.00000 0.00000 1.96468 0.00000
>>>>>>> 0.00000 0.00000 1.96468 0.00000 0.00000 0.00000
>>>>>>> 0.19612 0.00000 0.00000 0.00000 0.00000 0.00000
>>>>>>> 0.00000 0.19612 0.00000 0.00000 0.00000 0.00000
>>>>>>> 0.00000 0.00000 0.00000 0.00000 0.00000 0.19612
>>>>>>>
>>>>>>> b =
>>>>>>>
>>>>>>> Diagonal Matrix
>>>>>>>
>>>>>>> 3923.1 0 0 0 0 0
>>>>>>> 0 3923.1 0 0 0 0
>>>>>>> 0 0 3937.2 0 0 0
>>>>>>> 0 0 0 3937.2 0 0
>>>>>>> 0 0 0 0 3937.2 0
>>>>>>> 0 0 0 0 0 7846.2
>>>>>>>
>>>>>>>
>>>>>>> Numpy Results
>>>>>>>
>>>>>>> >>> STIFM
>>>>>>> array([[ 1020., 0., 0., 0., 0., 0.],
>>>>>>> [ 0., 1020., 0., 0., 0., 0.],
>>>>>>> [ 0., 0., 1020., 0., 0., 0.],
>>>>>>> [ 0., 0., 0., 102000., 0., 0.],
>>>>>>> [ 0., 0., 0., 0., 102000., 0.],
>>>>>>> [ 0., 0., 0., 0., 0., 204000.]])
>>>>>>>
>>>>>>> >>> MASSM
>>>>>>>
>>>>>>> array([[ 0.25907, 0. , 0. , 0. , 0. , 0.
>>>>>>> ],
>>>>>>> [ 0. , 0.25907, 0. , 0. , 0. , 0.
>>>>>>> ],
>>>>>>> [ 0. , 0. , 0.25907, 0. , 0. , 0.
>>>>>>> ],
>>>>>>> [ 0. , 0. , 0. , 26. , 0. , 0.
>>>>>>> ],
>>>>>>> [ 0. , 0. , 0. , 0. , 26. , 0.
>>>>>>> ],
>>>>>>> [ 0. , 0. , 0. , 0. , 0. , 26.
>>>>>>> ]])
>>>>>>>
>>>>>>> >>> a, b = linalg.eig(dot( linalg.pinv(MASSM), STIFM))
>>>>>>>
>>>>>>> >>> a
>>>>>>>
>>>>>>> array([ 3937.15984097, 3937.15984097, 3937.15984097,
>>>>>>> 3923.07692308,
>>>>>>> 3923.07692308, 7846.15384615])
>>>>>>>
>>>>>>> >>> b
>>>>>>>
>>>>>>> array([[ 1., 0., 0., 0., 0., 0.],
>>>>>>> [ 0., 1., 0., 0., 0., 0.],
>>>>>>> [ 0., 0., 1., 0., 0., 0.],
>>>>>>> [ 0., 0., 0., 1., 0., 0.],
>>>>>>> [ 0., 0., 0., 0., 1., 0.],
>>>>>>> [ 0., 0., 0., 0., 0., 1.]])
>>>>>>>
>>>>>>> On Tue, Dec 20, 2011 at 8:40 PM, Olivier Delalleau <shish at keba.be>wrote:
>>>>>>>
>>>>>>>> Hmm... ok ;) (sorry, I can't follow you there)
>>>>>>>>
>>>>>>>> Anyway, what kind of non-normalization are you after? I looked at
>>>>>>>> the doc for Matlab and it just says eigenvectors are not normalized,
>>>>>>>> without additional details... so it looks like it could be anything.
>>>>>>>>
>>>>>>>>
>>>>>>>> -=- Olivier
>>>>>>>>
>>>>>>>> 2011/12/20 Fahreddın Basegmez <mangabasi at gmail.com>
>>>>>>>>
>>>>>>>>> I am computing normal-mode frequency response of a mass-spring
>>>>>>>>> system. The algorithm I am using requires it.
>>>>>>>>>
>>>>>>>>> On Tue, Dec 20, 2011 at 8:10 PM, Olivier Delalleau <shish at keba.be>wrote:
>>>>>>>>>
>>>>>>>>>> I'm probably missing something, but... Why would you want
>>>>>>>>>> non-normalized eigenvectors?
>>>>>>>>>>
>>>>>>>>>> -=- Olivier
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> 2011/12/20 Fahreddın Basegmez <mangabasi at gmail.com>
>>>>>>>>>>
>>>>>>>>>>> Howdy,
>>>>>>>>>>>
>>>>>>>>>>> Is it possible to get non-normalized eigenvectors from
>>>>>>>>>>> scipy.linalg.eig(a, b)? Preferably just by using numpy.
>>>>>>>>>>>
>>>>>>>>>>> BTW, Matlab/Octave provides this with its eig(a, b) function but
>>>>>>>>>>> I would like to use numpy for obvious reasons.
>>>>>>>>>>>
>>>>>>>>>>> Regards,
>>>>>>>>>>>
>>>>>>>>>>> Fahri
>>>>>>>>>>>
>>>>>>>>>>
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>>>>>>>
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