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
>
> 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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