Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian

*To*: mathgroup at smc.vnet.net*Subject*: [mg86290] Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian*From*: David Reiss <dbreiss at gmail.com>*Date*: Fri, 7 Mar 2008 02:32:47 -0500 (EST)*References*: <fqb9ai$na0$1@smc.vnet.net> <200803021856.NAA19943@smc.vnet.net>

At the expense of stating the obvious, I suspect that most readers of this newsgroup understand this point... -_David On Mar 6, 3:09 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote: > Yes, since it uses quite different algorithm. Generally algorithms > used for numerical solution of equations or various matrix > computations are quite different from (and much faster than) those > used in the symbolic case. > > Andrzej Kozlowksi > > On 5 Mar 2008, at 09:38, David Reiss wrote: > > > 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}}

**Follow-Ups**:**Re: Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**References**:**Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian***From:*David Reiss <dbreiss@gmail.com>