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MathGroup Archive 2001

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Re: Eigenvalue Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg31999] Re: [mg31996] Eigenvalue Problem
  • From: BobHanlon at aol.com
  • Date: Mon, 17 Dec 2001 06:01:29 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

In a message dated 12/16/01 4:01:32 AM, smitsky at mindspring.com writes:

>Hi, could someone show me how to use Mathematica to sucessfully answer
>the
>following problem:
>
>Show that the Eigenvalues of A =
>
>[a][b]
>[c][d]
>
>must be real numbers. Thanks, Steve
>

A = {{a,b},{c,d}};

{\[Lambda], \[Mu]} = FullSimplify[Eigenvalues[A]]

{(1/2)*(a + d - Sqrt[(a - d)^2 + 
      4*b*c]), (1/2)*(a + d + 
    Sqrt[(a - d)^2 + 4*b*c])}

{u,v} = FullSimplify[Eigenvectors[A]]

{{-((-a + d + Sqrt[(a - d)^2 + 
        4*b*c])/(2*c)), 1}, 
  {(a - d + Sqrt[(a - d)^2 + 
       4*b*c])/(2*c), 1}}

Simplify[{A.u-\[Lambda]*u,A.v-\[Mu]*v}]

{{0, 0}, {0, 0}}

For the elements of the eigensystem to be real requires

(a - d)^2 + 4*b*c >= 0


Bob Hanlon
Chantilly, VA  USA


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