Eigenvector with constraint

• To: mathgroup at smc.vnet.net
• Subject: [mg32671] Eigenvector with constraint
• From: "Johannes Ludsteck" <johannes.ludsteck at wiwi.uni-regensburg.de>
• Date: Wed, 6 Feb 2002 03:41:22 -0500 (EST)
• Organization: Universitaet Regensburg
• Sender: owner-wri-mathgroup at wolfram.com

```Dear Mathgroup members,

I am searching for a eigenvector (for example) of the matrix

m = {{0.7,0.,0.},{0.3,0.5,0.9},{0.,0.5,0.1}}

associated with a unit eigenvalue, i.e want the
solution {e1,e2,e3} of the equation

{e1,e2,e3}==m.{e1,e2,e3}

Note: the columns of m sum to 1.
Since e1,e2 and e3 are probabilities, they must sum to 1.

When I apply FixedPoint,

FixedPoint[m.#&,{0,0,1}],

I obtain the solution {0.,0.642857,0.357143}.
Its elements sum to unity.

However, the eigenvector associated with eigenvalue 1 which I obtain
by typing

Eigensystem[p]
is {0.,-0.874157,-0.485643}.
It contains negative elements and thus cannot be probabilities.
How can I apply the restriction that e1,e2,e3>0 and e1+e2+e3 == 1?

Probably the reason for my question is less a problem of Mathematica,
but a defect in my mathematical background. Therefore please be
lenient with my 'stupid' question.

<><><><><><><><><><><><><><><><><><>
Johannes Ludsteck
Institut fuer Volkswirtschaftslehre
Lehrstuhl Prof. Dr. Moeller
Universitaet Regensburg
Universitaetsstrasse 31
93053 Regensburg
Tel +49/0941/943-2741

```

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