Example 6: The Jordan Normal Form

Take for example this 3x3 matrix, A



The eigenvalues have be found as , and
The eigenvector corresponding to the eigenvalue is

The only corresponding eigenvector to the eigenvalue is


A generalised eigenvector is found by calculating the kernel of

if we look at the above we have that


if and only if

A generalised eigenvector corresponding to where and gives

So the Jordan normal form can be transformed with a regular matrix P which is given by :

The Jordan Normal form of matrix A is given by

Checking that P-1AP=J holds:

as required