       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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