Re: Re: Eigenvalue Problem

• To: mathgroup at smc.vnet.net
• Subject: [mg17049] Re: [mg16991] Re: Eigenvalue Problem
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Wed, 14 Apr 1999 02:11:59 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```I do not think this problem is essentially to do with roots of equations.
Or rather, there is a deeper underlying problem. It is Mathematica's
inability to recognize certain complex expressions as real. To see this
one can take the following simple example:

Let

p=Sqrt[1-Sqrt[-a^2-1]]+Sqrt[1+Sqrt[-a^2-1]];

This is real for every real a, since it is then the sum of two conjugate
expressions. The reason is that the expressions -a^2-1 is always
negative. In fact it is easy to show by hand that

p=(2+a^2)^(1/4)*2*Cos[ArcCos[1/Sqrt[2+a^2]]/2]

However Mathematica can't do this (Try ComplexExpand) I think it should
be able to, since it has all the tools it needs at its disposal (the
Algebra`AlgebraicInequalities` package) but it can't. Is it really so
difficult to modify ComplexExpand to deal with such cases?

Some people responding to this posting mentioned the <<Miscellaneous`Real
Only` package as a solution. It seems to me that this package is meant
only for those who have not yet met complex numbers and is  useless
otherwise. If anyone thinks it can be helpful in this case he should try it.

On Sat, Apr 10, 1999, Paul Abbott <paul at physics.uwa.edu.au> wrote:

>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[3] A (80 A  + B  - 5 C )
>      - ArcCos[-(-------------------------------)]]) / Sqrt[3]
>      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
>____________________________________________________________________

Andrzej Kozlowski
Toyama International University
JAPAN
http://sigma.tuins.ac.jp/
http://eri2.tuins.ac.jp/

```

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