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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