Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian matrix

• To: mathgroup at smc.vnet.net
• Subject: [mg86124] Re: Mathematica 6 obtains imaginary eigenvalues for a Hermitian matrix
• From: David Bailey <dave at Remove_Thisdbailey.co.uk>
• Date: Sun, 2 Mar 2008 13:58:05 -0500 (EST)
• References: <fqb9ai\$na0\$1@smc.vnet.net>

```Sebastian Meznaric 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.
>
I am guessing here, but if you display the matrix it contains terms like
Sqrt[3] which has two possible answers +1.73.. and -1.73.... Although
this will not matter if you apply N to the matrix first, and then take
the eigenvalues, if you start by taking eigenvalues of the matrix you
get a horrendous expression involving Root objects (try it) and perhaps
the above confusion has arisen in that process.

In any case, for both efficiency and accuracy reasons it is preferable
to use use N as early as possible - EigenValues[N[matrix]]

David Bailey
http://www.dbaileyconsultancy.co.uk

```

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