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MathGroup Archive 2002

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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: [mg34988] Re: [mg34976] How to compute a MatrixPower using: A^n = P D^n Inverse[P]
  • From: Murray Eisenberg <murraye at attbi.com>
  • Date: Tue, 18 Jun 2002 02:48:28 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200206170726.DAA18874@smc.vnet.net>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

You do it in Mathematica just like the theory says to do it!  That means
that you construct P from the eigenvectors of A and the diagonal matrix
from the corresponding eigenvalues of A.  The essential Mathematica
function is Eigensystem.  Here's a "toy" example, using exact
arithmetic.  (With floating point arithmetic, of course, your results
may or may not be as good as you would wish).  

   A = {{2/5, 3/10}, {-13/40, 6/5}};

   esys = Eigensystem[A]
{{11/20, 21/20}, {{2, 1}, {6/13, 1}}}

   evecs = Last[esys]; evals = First[esys];
   P = Transpose[evecs] ;  Pinv = Inverse[P];
   diag = DiagonalMatrix[evals];
   
   A == P . diag . Pinv  (* theory says they're the same *)
True

   a[n_] := P . DiagonalMatrix[evals^n] . Pinv

   a[9] == MatrixPower[A, 9]
True


"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

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.       
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street
Amherst, MA 01375


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