Example 6.  Find the eigenvalues and eigenvectors of the matrix  .  

Solution 6.

Find the characteristic polynomial and the eigenvalues.

Investigate the eigen-pair  

Introduce the free variables and find the eigenvector.

In this case the eigenvector will have a nicer appearance if we replace t with 2t.

Verify the eigenpair.

Investigate the eigen-pair  

Introduce the free variables and find the eigenvector.

In this case the eigenvector will have a nicer appearance if we replace t with 2t.

Verify the eigenpair.

Investigate the eigen-pair  

Introduce the free variables and find the eigenvector.

In this case the eigenvector will have a nicer appearance if we replace t with 2t.

Verify the eigenpair.

Investigate the eigen-pair  

Introduce the free variables and find the eigenvector.

In this case the eigenvector will have a nicer appearance if we replace t with 2t.

Verify the eigenpair.

The four eigen-pairs are:

We can compare this with the results obtained using Mathematicas Eigensystem procedure.

(c) John H. Mathews 2004