Example 2.  Use the power method to find the dominant eigenvalue and eigenvector for the matrix  .  

Solution 2.

For illustration purposes we will set the maximum number of iterations to be 50 and .

That is    close to the dominant eigenvalue    and corresponding eigenvector  .  

Now check our work.

Compare with Mathematica's Eigensystem procedure.  Observe that Mathematica returns unit length eigenvectors.

Notice.  The numerical eigenvector found by Mathematica is which is a multiple of the the eigenvector    found by the power method, i.e.

        

Compare with Mathematica's Eigensystem procedure.  Use rational arithmetic.  This time Mathematica does not return a unit length eigenvector.

Aside.  Notice that this time the eigenvector found is  ,  which is a multiple of the vector    that was found with the power method.  

Caveat.  The accuracy of the eigenvalue and eigenvector we computed with the power method was limited by the number of iterations.

(c) John H. Mathews 2004