Finding Eigenvalues

We've spent most of our time so far talking about eigenvectors and now we turn our attention to eigenvalues. Let's begin by seeing how we can find the eigenvalues associated with an nxn matrix A.

Finding eigenvalues geometrically

Recall that the eigenvalue-eigenvector equation is Ax = rx. If this equation is true for a particular value of r and particular vector x then we say that r and x are associated or that they correspond to each other.

Given an eigenvector x of A, we can use this equation to find r by computing the norm of both sides

||Ax|| = |r| ||x||

and solving for r. This gives

|r| = ||Ax|| / ||x||

If the vector x is normalized then ||x|| = 1 so things are even simpler: |r| = ||Ax||.

Thus, the absolute value (magnitude) of the eigenvalue associated with the eigenvector x is given by ||Ax||.

Using the interactive tool, find the eigenvalues of the following matrices:

1.

 2 1
 1 2
2 and 2 1 and 3
2.

 2 1
 0 2
0 and 2 2
3.

 -1 0
 1 2
-1 and 2 1 and 2
4.

 2 1
 -1 2
-2 and 1 huh?