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Re: Eigenvalue Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg16897] Re: Eigenvalue Problem
  • From: bruck at pacificnet.net (Ronald Bruck)
  • Date: Tue, 6 Apr 1999 01:27:29 -0400
  • Organization: University of Southern California
  • References: <7e9odu$3km@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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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