Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian

• To: mathgroup at smc.vnet.net
• Subject: [mg86115] Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian
• From: David Reiss <dbreiss at gmail.com>
• Date: Sun, 2 Mar 2008 13:56:23 -0500 (EST)
• References: <fqb9ai\$na0\$1@smc.vnet.net>

```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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