       Re: Eigenvalue Problem

• To: mathgroup at smc.vnet.net
• Subject: [mg16991] Re: Eigenvalue Problem
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sat, 10 Apr 1999 02:13:30 -0400
• Organization: University of Western Australia
• References: <7e9odu\$3km@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Peter Huesser wrote:

> I am trying to solve the eigenvalue problem for the following matrix:
>
> m =     {{10 A, 0, B, 0, 0, 0},
>           {0, -2 A, 0, C, 0, 0},
>           {B, 0, -8 A, 0, C, 0},
>           {0, C, 0, -8 A, 0, B},
>           {0, 0, C, 0, -2 A, 0},
>           {0, 0, 0, B, 0, 10 A}}
>
> which is symmetric. Now mathematica returns some complex eigenvalues
> which is not
> possible for a real, symmetric matrix. Can anybody help me ? Maybe the
> error occurs because
> mathematica means that the coefficients are complex but how can I make
> them real ?

It can be shown that the (repeated) eigenvalues of this matrix can be
expressed in the explicitly real form,

2    2    2      1
(2 Sqrt[84 A  + B  + C ] Cos[- Pi (2 n - 1) +
3

2    2      2
1          3 Sqrt A (80 A  + B  - 5 C )
- ArcCos[-(-------------------------------)]]) / Sqrt
3                    2    2    2 3/2
(84 A  + B  + C )

where n=1,2,3.

For a discussion of this problem, see

ftp://ftp.physics.uwa.edu.au/pub/Mathematica/MathGroup/RealRoots.nb

Cheers,
Paul

____________________________________________________________________
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA  6907                     mailto:paul at physics.uwa.edu.au
AUSTRALIA                        http://www.physics.uwa.edu.au/~paul

God IS a weakly left-handed dice player
____________________________________________________________________

```

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