Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian
- To: mathgroup at smc.vnet.net
- Subject: [mg86179] Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian
- From: David Reiss <dbreiss at gmail.com>
- Date: Wed, 5 Mar 2008 03:38:35 -0500 (EST)
- References: <fqb9ai$na0$1@smc.vnet.net> <200803021856.NAA19943@smc.vnet.net>
Thanks.... I stand (or sit) corrected. I note that although N[Eigenvalues[matrix]] does not give correct results (with the bug that Daniel Lichtblau explains), Eigenvalues[N[matrix]] does indeed work as expected... --David On Mar 3, 4:48 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote: > 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 am, 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 - > >> 0.88758 \[ImaginaryI], -7.37965 + 2.32729 \[ImaginaryI], > >> -7.37965 - > >> 2.32729 \[ImaginaryI], -4.46655 + 2.59738 \[ImaginaryI], > >> -4.46655 - > >> 2.59738 \[ImaginaryI], 4.36971, 3.21081, -2.32456 + > >> 2.10914 \[ImaginaryI], -2.32456 - 2.10914 \[ImaginaryI], > >> 2.04366+ 0.552265 \[ImaginaryI], > >> 2.04366- 0.552265 \[ImaginaryI], -0.249588 + > >> 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, > >> 0, -Sqrt[3], 0, 0, Sqrt[3], 0, 0, 0, 0, 0, 0, 0}, {-Sqrt[3], 0, = > >> -4, > >> 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), 2 Sqrt[2/3], 0, 0, Sqrt[3], = > >> 0, > >> 0, 0, 0, 0, 0}, {0, -Sqrt[3], > >> 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -4/3, -(2 Sqrt[2])/3, 0, 0, = > >> 0, > >> Sqrt[3], 0, 0, 0, 0, 0}, {0, 0, 2 Sqrt[2/3], -(2 Sqrt[2])/3, > >> 7/3, 0, > >> 0, 0, 0, Sqrt[3], 0, 0, 0, 0}, {Sqrt[3], 0, 0, 0, 0, -4, 0, > >> 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, 0, 2 Sqrt[2/3], 0, 0, 0}, {0, > >> Sqrt[3], 0, 0, 0, 0, -4, 0, 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, 0,= > >> 2 Sqrt[2/3], 0, 0}, {0, 0, Sqrt[3], 0, 0, > >> 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, -14/3, > >> 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), 2 Sqrt[2/3], (2 Sqrt[2])/3, = > >> 0, > >> 0, 0}, {0, 0, 0, Sqrt[3], 0, 0, 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), > >> 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -2, -(2 Sqrt[2])/3, 0, ( > >> 2 Sqrt[2])/3, 0, 0}, {0, 0, 0, 0, Sqrt[3], 0, 0, > >> 2 Sqrt[2/3], -(2 Sqrt[2])/3, -7/3, 0, 0, > >> 2 (1/(3 Sqrt[2]) + (2 Sqrt[2])/3), Sqrt[10/3]}, {0, 0, 0, 0, 0, > >> 2 Sqrt[2/3], 0, (2 Sqrt[2])/3, 0, 0, -16/3, > >> 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), 2 Sqrt[2/3], 0}, {0, 0, 0, = > >> 0, 0, > >> 0, 2 Sqrt[2/3], 0, (2 Sqrt[2])/3, 0, > >> 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -8/3, -(2 Sqrt[2])/3, 0}, = > >> {0, 0, > >> 0, 0, 0, 0, 0, 0, 0, 2 (1/(3 Sqrt[2]) + (2 Sqrt[2])/3), > >> 2 Sqrt[2/3], -(2 Sqrt[2])/3, 1/2, > >> 2 (-Sqrt[5/3]/16 - Sqrt[15]/16)}, {0, 0, 0, 0, 0, 0, 0, 0, 0, = > >> Sqrt[ > >> 10/3], 0, 0, 2 (-Sqrt[5/3]/16 - Sqrt[15]/16), 7/2}}
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- Re: Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian
- From: Andrzej Kozlowski <akoz@mimuw.edu.pl>
- Re: Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian
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- Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian
- From: David Reiss <dbreiss@gmail.com>
- Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian