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Re: How to compute a MatrixPower using: A^n = P D^n Inverse[P]

• To: mathgroup at smc.vnet.net
• Subject: [mg34994] Re: [mg34976] How to compute a MatrixPower using: A^n = P D^n Inverse[P]
• From: Adriano Pascoletti <pascolet at dimi.uniud.it>
• Date: Tue, 18 Jun 2002 02:48:37 -0400 (EDT)
• References: <200206170726.DAA18874@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

You can use Eigensystem or JordanDecomposition as follows:

Eigensystem

In[1]:=
A = {{3, 1, 0}, {1, 2, -1}, {0, -1, 3}};
{evals, M} = Eigensystem[A]; M = Transpose[M];
   M . DiagonalMatrix[evals^n] . Inverse[M]

Out[2]=
{{1/6 + 3^n/2 + 4^n/3, -(1/3) + 4^n/3,
    -(1/6) + 3^n/2 - 4^n/3}, {-(1/3) + 4^n/3, 2/3 + 4^n/3,
    1/3 - 4^n/3}, {-(1/6) + 3^n/2 - 4^n/3, 1/3 - 4^n/3,
    1/6 + 3^n/2 + 4^n/3}}

JordanDecomposition

In[3]:=
{P, Diag} = JordanDecomposition[A]; evals = Tr[Diag, List];
P . DiagonalMatrix[evals^n] . Inverse[P]
Out[4]=
{{1/6 + 3^n/2 + 4^n/3, -(1/3) + 4^n/3,
    -(1/6) + 3^n/2 - 4^n/3}, {-(1/3) + 4^n/3, 2/3 + 4^n/3,
    1/3 - 4^n/3}, {-(1/6) + 3^n/2 - 4^n/3, 1/3 - 4^n/3,
    1/6 + 3^n/2 + 4^n/3}}

Adriano Pascoletti

>I have the matrix
>
>A == {{3,1,0},{1,2,-1},{0,-1,3}}
>
>For educational purpose I would like to evaluate
>
>A^n (* I mean MatrixPower[A,n]*)
>
>using the following matrix property
>
>A^n == P D^n Inverse[P]  (*D mean Diagonal Matrix *)
>
>How can I do with Mathematica? (Methods to obtain P and D)
>
>Thanks
>
>Guillermo Sanchez

• References:
  • How to compute a MatrixPower using: A^n = P D^n Inverse[P]
    • From: "J. Guillermo Sanchez" <guillerm@usal.es>