Example 3.  Find the eigenvalues and eigenvectors of the matrix  .  

Solution 3.

Find the characteristic polynomial and the eigenvalues.

Let us plot and see where the roots are located

Although this example has been "cooked up" so that the values are simple, we should be aware that a root finding method could be employed to find the eigenvalues.  For illustration, we can use the Newton-Raphson method.

Since we have solved for roots in previous modules, we will concentrate our effort on solving for the eigenvectors.
First, we shall automate the procedure for finding the roots of the characteristic polynomial, which is one way to find the eigenvalues.

Investigate the eigen-pair  

Introduce the free variables and find the eigenvector.

Verify the eigenpair.

Investigate the eigen-pair  

Introduce the free variables and find the eigenvector.

Verify the eigenpair.

Investigate the eigen-pair  

Introduce the free variables and find the eigenvector.

Verify the eigenpair.

The three eigen-pairs are:

We can compare this with the results obtained using Mathematica's Eigensystem procedure.

(c) John H. Mathews 2004