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 > > > >