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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*: [mg34987] Re: [mg34976] How to compute a MatrixPower using: A^n = P D^n Inverse[P]
*From*: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
*Date*: Tue, 18 Jun 2002 02:48:27 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Since your matrix is symmetric and has distinct eigenvalues this is
rather easy.
Let's just do one case, say n=3.
First let's compute:
In[1]:=
A={{3,1,0},{1,2,-1},{0,-1,3}};
In[2]:=
B=MatrixPower[A,3];
Next we find the eigenvalues and eigenvectors of B.
In[3]:=
{eigenvals, eigenvecs} = Eigensystem[B];
We normalize the eigenvectors and compute the change of basis matrix:
In[4]:=
q = Map[1/Sqrt[#.#]&, eigenvecs]*eigenvecs;
the diagonal matrix has the eigenvalues along the diagonal:
In[5]:=
d=DiagonalMatrix[eigenvals];
Its transpose = its inverse
In[6]:=
Transpose[q]==Inverse[q]
Out[6]=
True
Finally we check the formula:
In[7]:=
B==Inverse[q].d.q
Out[7]=
True
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
On Monday, June 17, 2002, at 04:26 PM, J. Guillermo Sanchez wrote:
>
> 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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