Inicio del mensaje reenviado:
> De: Arnau Mir <arnau....@uib.es>
> Fecha: 14 de noviembre de 2011 13:24:31 GMT+01:00
> Para: Martin Maechler <maech...@stat.math.ethz.ch>
> Asunto: Re: [R] How to compute eigenvectors and eigenvalues?
>
> Sorry, but I can't explain very well.
>
>
> The matrix 4*mp is:
>
> 4*mp
> [,1] [,2] [,3]
> [1,] 0 1 1
> [2,] 3 0 1
> [3,] 1 3 2
> >
>
> The eigenvalues are:
> lam
> [1] 4 -1 -1
>
> There is only one eigenvector linearly independent with eigenvalue -1. This
> eigenvector is c(0,1,-1) but R computes two eigenvectors:
>
> > V
> [,1] [,2] [,3]
> [1,] 0.3207501 1.068531e-08 -1.068531e-08
> [2,] 0.4490502 -7.071068e-01 -7.071068e-01
> [3,] 0.8339504 7.071068e-01 7.071068e-01
>
>
> One is
> V[,2]
> [1] 1.068531e-08 -7.071068e-01 7.071068e-01
>
> and the other one:
>
> > V[,3]
> [1] -1.068531e-08 -7.071068e-01 7.071068e-01
>
> The first component of these two vectors must be zero but due to numerical
> errors, R computes 1.068531e-08 and -1.068531e-08 respectively.
>
> So, the matrix V is singular but R isn't aware of.
>
> You are right when you say:
>
> V %*% diag(lam) %*% solve(V)
> [,1] [,2] [,3]
> [1,] 8.788496e-09 1.000000e+00 1
> [2,] 3.000000e+00 -9.313226e-10 1
> [3,] 1.000000e+00 3.000000e+00 2
>
>
> but this expression doesn't make any sense because solve(V) doesn't exist.
>
> What I want is that R responses that mp or 4*mp is not diagonalizable and
> solve(V) doesn't exist.
>
> Arnau.
> El 14/11/2011, a las 13:06, Martin Maechler escribió:
>
>>
>>> Consider the following matrix:
>>
>>>> mp <- matrix(c(0,1/4,1/4,3/4,0,1/4,1/4,3/4,1/2),3,3,byrow=T)
>>
>>>> mp
>>> [,1] [,2] [,3]
>>> [1,] 0.00 0.25 0.25
>>> [2,] 0.75 0.00 0.25
>>> [3,] 0.25 0.75 0.50
>>
>>> The eigenvectors of the previous matrix are 1, 0.25 and 0.25 and it is not
>>> a diagonalizable matrix.
>>
>>> When you try to find the eigenvalues and eigenvectors with R, R responses:
>>
>>>> eigen(mp)
>>
>>> $values
>>> [1] 1.00 -0.25 -0.25
>>
>>> $vectors
>>> [,1] [,2] [,3]
>>> [1,] 0.3207501 1.068531e-08 -1.068531e-08
>>> [2,] 0.4490502 -7.071068e-01 -7.071068e-01
>>> [3,] 0.8339504 7.071068e-01 7.071068e-01
>>
>>> The eigenvalues are correct but the eigenvectors aren't.
>>
>> Well, let's look at 4*mp which is an integer matrix:
>>
>>> (M <- matrix(c(rep(c(0, 3, 1, 1), 2),2), 3,3)); em <- eigen(M); V <-
>>> em$vectors; lam <- em$values
>> [,1] [,2] [,3]
>> [1,] 0 1 1
>> [2,] 3 0 1
>> [3,] 1 3 2
>>> all.equal(M, V %*% diag(lam) %*% solve(V))
>> [1] TRUE
>>> zapsmall(V %*% diag(lam) %*% solve(V))
>> [,1] [,2] [,3]
>> [1,] 0 1 1
>> [2,] 3 0 1
>> [3,] 1 3 2
>>>
>>
>> So, as you see V is not singular,
>> and the basic property of the eigenvalue decomposition is
>> fulfilled:
>>
>> M = V Lambda V^{-1}
>> or
>> M V = V Lambda
>>
>> i.e. each column of V *is* eigenvector to the corresponding
>> eigenvalue.
>>
>>
>>> Moreover, if you try to compute the inverse of the matrix of eigenvectors,
>>> R is not aware that this matrix is singular:
>>
>>>> solve(eigen(mp)$vectors)
>>> [,1] [,2] [,3]
>>> [1,] 6.235383e-01 6.235383e-01 6.235383e-01
>>> [2,] 3.743456e+07 -9.358641e+06 -9.358640e+06
>>> [3,] -3.743456e+07 9.358640e+06 9.358641e+06
>>
>> it isn't...
>>
>>
>>
>>> My question is: how can I fix it?
>>
>> No need to fix anything, as nothing is broken ;-)
>>
>> Martin Maechler, ETH Zurich
>>
>
> ------------------------------------------------------------
> Arnau Mir Torres
> Edifici A. Turmeda
> Campus UIB
> Ctra. Valldemossa, km. 7,5
> 07122 Palma de Mca.
> tel: (+34) 971172987
> fax: (+34) 971173003
> email: arnau....@uib.es
> URL: http://dmi.uib.es/~arnau
> ------------------------------------------------------------
>
------------------------------------------------------------
Arnau Mir Torres
Edifici A. Turmeda
Campus UIB
Ctra. Valldemossa, km. 7,5
07122 Palma de Mca.
tel: (+34) 971172987
fax: (+34) 971173003
email: arnau....@uib.es
URL: http://dmi.uib.es/~arnau
------------------------------------------------------------
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