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

• To: mathgroup at smc.vnet.net
• Subject: [mg34985] RE: [mg34976] How to compute a MatrixPower using: A^n = P D^n Inverse[P]
• From: "Florian Jaccard" <jaccardf at eicn.ch>
• Date: Tue, 18 Jun 2002 02:48:24 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hello !

Here a way like you want the students to do it :

In[3]:=
a={{3,1,0},{1,2,-1},{0,-1,3}}

Out[3]=
{{3,1,0},{1,2,-1},{0,-1,3}}

matrice of base change :

In[4]:=
p=Transpose[Eigenvectors[a]]

Out[4]=
{{-1,1,-1},{2,0,-1},{1,1,1}}

diagonal matrix :

In[5]:=
d=Inverse[p].a.p

Out[5]=
{{1,0,0},{0,3,0},{0,0,4}}

contrôle :

In[6]:=
Eigenvalues[a]

Out[6]=
{1,3,4}

In[9]:=
Simplify[aPuissancen=p.d^n.Inverse[p],n>0]//MatrixForm

contrôle :

In[12]:=
Simplify[aPuissancen\[Equal]MatrixPower[a,n],n>0]

Out[12]=
True

Meilleures salutations

Florian Jaccard
professeur de Mathématiques
EICN-HES
7, av. de l'Hôtel-de-Ville
CH-2400 Le Locle
e-mail : jaccardf at eicn.ch

-----Message d'origine-----
De : J. Guillermo Sanchez [mailto:guillerm at usal.es]
Envoyé : lun., 17. juin 2002 09:27
À : mathgroup at smc.vnet.net
Objet : [mg34976] How to compute a MatrixPower using: A^n = P D^n
Inverse[P]

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

```

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