       Re: Eigenvalue Problem

• To: mathgroup at smc.vnet.net
• Subject: [mg16899] Re: [mg16894] Eigenvalue Problem
• From: Daniel Lichtblau <danl>
• Date: Tue, 6 Apr 1999 01:27:30 -0400
• References: <199904050624.CAA03551@smc.vnet.net.>
• Sender: owner-wri-mathgroup at wolfram.com

```Peter Haesser 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 eigenvalues are symbolic roots of a polynomial. As such they are
neither real nor complex valued unless and until one gives values for
the parameters {A,B,C}. So it is unclear what you mean when you say
Mathematica has returned complex eigenvalues.

One possiblity is that you would prefer a result in terms of Root[]
objects rather than radicals. This may be obtained as follows.

In:= SetOptions[Roots,Quartics->False, Cubics->False];

In:= InputForm[eigs2 = Eigenvalues[m]]

Out//InputForm=
{Root[-160*A^3 - 2*A*B^2 + 10*A*C^2 - 84*A^2*#1 - B^2*#1 - C^2*#1 + #1^3
& ,
1], Root[-160*A^3 - 2*A*B^2 + 10*A*C^2 - 84*A^2*#1 - B^2*#1 - C^2*#1 +
#1^3 & , 1], Root[-160*A^3 - 2*A*B^2 + 10*A*C^2 - 84*A^2*#1 - B^2*#1
-
C^2*#1 + #1^3 & , 2],
Root[-160*A^3 - 2*A*B^2 + 10*A*C^2 - 84*A^2*#1 - B^2*#1 - C^2*#1 + #1^3
& ,
2], Root[-160*A^3 - 2*A*B^2 + 10*A*C^2 - 84*A^2*#1 - B^2*#1 - C^2*#1 +
#1^3 & , 3], Root[-160*A^3 - 2*A*B^2 + 10*A*C^2 - 84*A^2*#1 - B^2*#1
-
C^2*#1 + #1^3 & , 3]}

Daniel Lichtblau
Wolfram Research

```

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