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Re: Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian

  • To: mathgroup at smc.vnet.net
  • Subject: [mg86142] Re: [mg86115] Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 3 Mar 2008 04:40:54 -0500 (EST)
  • References: <fqb9ai$na0$1@smc.vnet.net> <200803021856.NAA19943@smc.vnet.net>

Eigenvalues or characteristic values of a  are defined (or rather, can  
be defined) as the roots of the characteristic polynomial - and it  
does not matter is the matrix is invertible or not. Indeed, for a  
nilpotent matrix, such as

M = {{-1, I}, {I, 1}}

we have

In[39]:= Eigenvalues[M]
Out[39]= {0, 0}


and

In[40]:= CharacteristicPolynomial[M, x]
Out[40]= x^2

Moreover, the problem has nothing to do with numerical precision  
because in this case the exact eigenvalues do not satisfy the  
characteristic polynomial and in fact are the exact roots of a  
different polynomial of the same degree (as shown in my first post in  
this thread). Very weird.

Andrzej Kozlowski


On 2 Mar 2008, at 19:56, David Reiss wrote:

> Note that your matrix is not invertible (its determinant is zero).  So
> this is the source of your problem...
>
> Hope that this helps...
>
> -David
> A WorkLife FrameWork
> E x t e n d i n g MATHEMATICA's Reach...
> http://scientificarts.com/worklife/
>
>
>
>
> On Mar 1, 4:57=A0am, Sebastian Meznaric <mezna... at gmail.com> wrote:
>> I have a 14x14 Hermitian matrix, posted at the bottom of this  
>> message.
>> The eigenvalues that Mathematica obtains using the
>> N[Eigenvalues[matrix]] include non-real numbers:
>> {-9.41358 + 0.88758 \[ImaginaryI], -9.41358 -
>> =A0 0.88758 \[ImaginaryI], -7.37965 + 2.32729 \[ImaginaryI],  
>> -7.37965 -
>> =A0 2.32729 \[ImaginaryI], -4.46655 + 2.59738 \[ImaginaryI],  
>> -4.46655 -
>> =A0 2.59738 \[ImaginaryI], 4.36971, 3.21081, -2.32456 +
>> =A0 2.10914 \[ImaginaryI], -2.32456 - 2.10914 \[ImaginaryI],
>> =A02.04366+ 0.552265 \[ImaginaryI],
>> =A02.04366- 0.552265 \[ImaginaryI], -0.249588 +
>> =A0 1.29034 \[ImaginaryI], -0.249588 - 1.29034 \[ImaginaryI]}.
>> However, if you do Eigenvalues[N[matrix]] it obtains different  
>> results
>> {-9.09122, -7.41855, -7.41855, -7.2915, 4.33734, -4., -4., 3.2915, \
>> -3.24612, -2.38787, -2.38787, 1.80642, 1.80642, 0}.
>>
>> These results agree with Solve[CharacteristicPolynomial[matrix,x],x].
>> Therefore I assume that the latter are correct. Has anyone seen this?
>> I am using 6.0.0.
>>
>> Here is the matrix:
>> {{-6, 0, -Sqrt[3], 0, 0, Sqrt[3], 0, 0, 0, 0, 0, 0, 0, 0}, {0, -6,
>> =A0 0, -Sqrt[3], 0, 0, Sqrt[3], 0, 0, 0, 0, 0, 0, 0}, {-Sqrt[3], 0,  
>> -4,
>> =A0 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), 2 Sqrt[2/3], 0, 0, Sqrt[3],  
>> 0,
>> =A0 0, 0, 0, 0, 0}, {0, -Sqrt[3],
>> =A0 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -4/3, -(2 Sqrt[2])/3, 0, 0,  
>> 0,
>> =A0 Sqrt[3], 0, 0, 0, 0, 0}, {0, 0, 2 Sqrt[2/3], -(2 Sqrt[2])/3,  
>> 7/3, 0,
>> =A0 =A00, 0, 0, Sqrt[3], 0, 0, 0, 0}, {Sqrt[3], 0, 0, 0, 0, -4, 0,
>> =A0 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, 0, 2 Sqrt[2/3], 0, 0, 0}, {0,
>> =A0 Sqrt[3], 0, 0, 0, 0, -4, 0, 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, 0,
>> =A0 2 Sqrt[2/3], 0, 0}, {0, 0, Sqrt[3], 0, 0,
>> =A0 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, -14/3,
>> =A0 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), 2 Sqrt[2/3], (2 Sqrt[2])/3,  
>> 0,
>> =A0 0, 0}, {0, 0, 0, Sqrt[3], 0, 0, 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4),
>> =A0 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -2, -(2 Sqrt[2])/3, 0, (
>> =A0 2 Sqrt[2])/3, 0, 0}, {0, 0, 0, 0, Sqrt[3], 0, 0,
>> =A0 2 Sqrt[2/3], -(2 Sqrt[2])/3, -7/3, 0, 0,
>> =A0 2 (1/(3 Sqrt[2]) + (2 Sqrt[2])/3), Sqrt[10/3]}, {0, 0, 0, 0, 0,
>> =A0 2 Sqrt[2/3], 0, (2 Sqrt[2])/3, 0, 0, -16/3,
>> =A0 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), 2 Sqrt[2/3], 0}, {0, 0, 0,  
>> 0, 0,
>> =A0 =A00, 2 Sqrt[2/3], 0, (2 Sqrt[2])/3, 0,
>> =A0 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -8/3, -(2 Sqrt[2])/3, 0},  
>> {0, 0,
>> =A0 =A00, 0, 0, 0, 0, 0, 0, 2 (1/(3 Sqrt[2]) + (2 Sqrt[2])/3),
>> =A0 2 Sqrt[2/3], -(2 Sqrt[2])/3, 1/2,
>> =A0 2 (-Sqrt[5/3]/16 - Sqrt[15]/16)}, {0, 0, 0, 0, 0, 0, 0, 0, 0,  
>> Sqrt[
>> =A0 10/3], 0, 0, 2 (-Sqrt[5/3]/16 - Sqrt[15]/16), 7/2}}
>
>



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