Example 11.  Find the eigenvalues of the following matrix  A.

Solution 11.

Which method do we trust ?  
Are the answers on the left better or are the answers on right better ?

We trust Mathematica's Eigenvalue subroutine !   

We should go with the answers on the right.  This is a transition matrix and it is known that one of its eigenvalues is  .   

The eigenvalue-eigenvector methods we study seem overbearing, but they are necessary when .

We could try to envision the difficulties of finding the root of

Clearly, there are computational problems in evaluating the Newton-Raphson iteration function.  This is why there are methods for finding eigenvalues that do not rely on finding roots of polynomials.  

(c) John H. Mathews 2004