Mathematica 6 obtains imaginary eigenvalues for a Hermitian matrix

• To: mathgroup at smc.vnet.net
• Subject: [mg86010] Mathematica 6 obtains imaginary eigenvalues for a Hermitian matrix
• From: Sebastian Meznaric <meznaric at gmail.com>
• Date: Sat, 1 Mar 2008 04:47:03 -0500 (EST)

```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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