Let's make a worked example of Jordan form calculation for a 3x3 matrix.
Let's the matrix
Calculate the roots of characteristic polynomial, ie calculate the eigenspace AX=λX, this is given for the equation system A-λI=0
Therefore, we have the λ=3 triple multiplicity eigenvalue.
We need to know the dimension of the eigenspace generated by this eigenvalue, ie, calculate dim [Ker (A-3I)], to do that, we solve the system
So, it appears
Thus we obtain a single eigenvector
Thus the dimension of the eigenspace is 1. Therefore, A is not diagonalizable, and
we know that the Jordan form of A is
We calculate the basis of eigenvectors, for now we only have one, the v1,
that is, we have to calculate two more eigenvectors. For this, we apply the theory,
ie, we calculate a vector, v2 such that.
(A-3I)v2 = v1, ie solve the system
Now we have a second eigenvector, we now calculate the third one,
v3, such that
(A-3I)v3 = v2, ie solve the system
We're done because we have to
A = PJP-1
And P is those matrix formed the eigenvectors placed as columns, ie