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Re: Eigenvalue Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg16976] Re: Eigenvalue Problem
  • From: Pierre infelta <pierre.infelta at epfl.ch>
  • Date: Sat, 10 Apr 1999 02:13:23 -0400
  • Organization: EPFL
  • References: <7e9odu$3km@smc.vnet.net> <7ef0c1$c4f$2@dragonfly.wolfram.com>
  • Sender: owner-wri-mathgroup at wolfram.com

Needs["Miscellaneous`RealOnly`"]

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}}

Eigenvalues[m]                       

The answer replaces the nonreal roots by "nonreal".

Ronald Bruck wrote:
> 
> In article <7e9odu$3km at smc.vnet.net>, Peter Huesser <phuesser at bluewin.ch> wrote:
> 
> > Hello everybody
> >
> > 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 ?
> >
> > Thank's in advance for any help.
> >
> >
> >     Peter Huesser
> 
> The problem is that det (xI - m) is the square of a cubic polynomial.
> Mathematica is using Cardano's formula, which often leads to complex
> numbers--which cancel out, leaving only real numbers.  This behavior CAN'T
> be overcome, restricting yourself to algebraic functions.  It is possible
> to do it using trig functions, and I think that's documented somewhere.
> It's appeared in this newsgroup before, certainly.
> 
> Try a web search on "cubic equation".
> 
> --Ron Bruck


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