Example 2.  Find the eigenvalues and eigenvectors of the matrix  .

Solution 2.

This is equivalent to the linear system

    

Set and solve for  , and get the eigenvector  

Verify the eigenpair.

This is equivalent to the linear system

    

Set and solve for  , and get the eigenvector  

Verify the eigenpair.

Remark. Newton's method can be used to find the roots of the characteristic polynomial.

For the first eigenvalue.

Which is an approximation to the eigenvalue

For the second eigenvalue.

Which is an approximation to the eigenvalue

(c) John H. Mathews 2004