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Re: Eigensystem consistency

  • To: mathgroup at smc.vnet.net
  • Subject: [mg84032] Re: [mg84001] Eigensystem consistency
  • From: DrMajorBob <drmajorbob at bigfoot.com>
  • Date: Fri, 7 Dec 2007 03:05:55 -0500 (EST)
  • References: <30219775.1196952663537.JavaMail.root@m35> <op.t2xd9zgcqu6oor@monster.gateway.2wire.net>
  • Reply-to: drmajorbob at bigfoot.com

Sorry, my earlier post was somewhat incorrect.

The difference isn't that N "supervises" the computations of Eigensystem 
in your first example; it does change the exact Eigensystem result to  
Real, as in

Eigensystem[{{\[Sigma]1^2, \[Rho] \[Sigma]1 \[Sigma]2}, {\[Rho] \
\[Sigma]1 \[Sigma]2, \[Sigma]2^2}} /. {\[Sigma]1 -> 1, \[Sigma]2 ->
     3, \[Rho] -> 98/100}]
N@%

{{1/50 (250 + Sqrt[61609]),
   1/50 (250 - Sqrt[61609])}, {{-(150/49) + 1/147 (250 + Sqrt[61609]),
    1}, {-(150/49) + 1/147 (250 - Sqrt[61609]), 1}}}

{{9.96423, 0.0357679}, {{0.32797, 1.}, {-3.04906, 1.}}}

Notice that Eigensystem gives an EXACT result, based on symbolic  
computations, as in

Eigensystem[{{\[Sigma]1^2, \[Rho] \[Sigma]1 \[Sigma]2}, {\[Rho] \
\[Sigma]1 \[Sigma]2, \[Sigma]2^2}}]

{{1/2 (\[Sigma]1^2 + \[Sigma]2^2 -
      Sqrt[\[Sigma]1^4 - 2 \[Sigma]1^2 \[Sigma]2^2 +
       4 \[Rho]^2 \[Sigma]1^2 \[Sigma]2^2 + \[Sigma]2^4]),
   1/2 (\[Sigma]1^2 + \[Sigma]2^2 +
      Sqrt[\[Sigma]1^4 - 2 \[Sigma]1^2 \[Sigma]2^2 +
       4 \[Rho]^2 \[Sigma]1^2 \[Sigma]2^2 + \[Sigma]2^4])}, {{-((-\
\[Sigma]1^2 + \[Sigma]2^2 +
      Sqrt[\[Sigma]1^4 - 2 \[Sigma]1^2 \[Sigma]2^2 +
       4 \[Rho]^2 \[Sigma]1^2 \[Sigma]2^2 + \[Sigma]2^4])/(
     2 \[Rho] \[Sigma]1 \[Sigma]2)),
    1}, {-((-\[Sigma]1^2 + \[Sigma]2^2 -
      Sqrt[\[Sigma]1^4 - 2 \[Sigma]1^2 \[Sigma]2^2 +
       4 \[Rho]^2 \[Sigma]1^2 \[Sigma]2^2 + \[Sigma]2^4])/(
     2 \[Rho] \[Sigma]1 \[Sigma]2)), 1}}}

These are also correct eigenvectors:

2 \[Rho] \[Sigma]1 \[Sigma]2 vectors

{{\[Sigma]1^2 - \[Sigma]2^2 -
    Sqrt[\[Sigma]1^4 - 2 \[Sigma]1^2 \[Sigma]2^2 +
     4 \[Rho]^2 \[Sigma]1^2 \[Sigma]2^2 + \[Sigma]2^4],
   2 \[Rho] \[Sigma]1 \[Sigma]2}, {\[Sigma]1^2 - \[Sigma]2^2 +
    Sqrt[\[Sigma]1^4 - 2 \[Sigma]1^2 \[Sigma]2^2 +
     4 \[Rho]^2 \[Sigma]1^2 \[Sigma]2^2 + \[Sigma]2^4],
   2 \[Rho] \[Sigma]1 \[Sigma]2}}

In your second example, the inputs are Real and computation is numeric, as  
in

{{\[Sigma]1^2, \[Rho] \[Sigma]1 \[Sigma]2}, {\[Rho] \[Sigma]1 \
\[Sigma]2, \[Sigma]2^2}} /.
  N[{\[Sigma]1 -> 1, \[Sigma]2 -> 3, \[Rho] -> 98/100}]
Eigensystem[%]

{{1., 2.94}, {2.94, 9.}}

{{9.96423, 0.0357679}, {{0.311638, 0.950201}, {0.950201, -0.311638}}}

As I said before, the two results ARE the same, except for scaling. If  
there were repeated eigenvalues, they need not be the same even to that 
degree, so long as the eigenvectors for a specified eigenvalue span the 
same space in both results.

Bobby

On Thu, 06 Dec 2007 12:14:13 -0600, DrMajorBob <drmajorbob at bigfoot.com>
wrote:

> Both results ARE "the same", to the extent that equivalent eigensystems  
> SHOULD be the same. That is, one set of eigenvectors is just a rescaled  
> version of the other. The reason this can occur is that N doesn't merely  
> evaluate its argument, then change it to a Real. It controls evaluation  
> of its argument in order to get a specific precision in the result. So 
> Eigensystem within N computes differently.
>
> Hence these are the same,
>
> Eigensystem[{{\[Sigma]1^2, \[Rho] \[Sigma]1 \[Sigma]2}, {\[Rho] \
> \[Sigma]1 \[Sigma]2, \[Sigma]2^2}} /.
>    N[{\[Sigma]1 -> 1, \[Sigma]2 -> 3, \[Rho] -> 98/100}]]
> N@%
>
> {{9.96423, 0.0357679}, {{0.311638, 0.950201}, {0.950201, -0.311638}}}
>
> {{9.96423, 0.0357679}, {{0.311638, 0.950201}, {0.950201, -0.311638}}}
>
> but your first expression is "different".
>
> Bobby
>
> On Thu, 06 Dec 2007 02:12:36 -0600, Arturas Acus <acus at itpa.lt> wrote:
>
>> Dear group,
>>
>> why these two calculations give different rezults?
>>
>>
>>
>> In[1]:= N[
>>  Eigensystem[{{\[Sigma]1^2, \[Rho] \[Sigma]1 \[Sigma]2}, {\[Rho] \
>> \[Sigma]1 \[Sigma]2, \[Sigma]2^2}} /. {\[Sigma]1 -> 1, \[Sigma]2 ->
>>      3, \[Rho] -> 98/100}]]
>>
>> Out[1]= {{9.96423, 0.0357679}, {{0.32797, 1.}, {-3.04906, 1.}}}
>>
>>
>>
>> and
>>
>> In[2]:= Eigensystem[{{\[Sigma]1^2, \[Rho] \[Sigma]1 \[Sigma]2}, {\
>> \[Rho] \[Sigma]1 \[Sigma]2, \[Sigma]2^2}} /.
>>   N[{\[Sigma]1 -> 1, \[Sigma]2 -> 3, \[Rho] -> 98/100}]]
>>
>> Out[2]= {{9.96423,
>>   0.0357679}, {{0.311638, 0.950201}, {0.950201, -0.311638}}}
>>
>>
>>
>>
>>
>>
>
>
>



-- 

DrMajorBob at bigfoot.com


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