- First thing we do is find out the spectrum of , that is, its eigenvalues. We shall mark them as . We do this however we can, usually by finding the zeros of the characteristic polynomial, that is solving the equation for
- In the next step we will be examining the sequence of generalized eigenspaces (i.e. for a given ) for each .
We calculate each kernel by simply solving the equation
Eventually, the two consecutive kernels in a sequence will be the same -
- When we have found the first such kernel, we take a look at the one that it succeeded - . We must now choose such a vector that is in but not in . For example if
whereas , we can choose
- We now consider a sequence:
Eventually one vector will be zero,
- Concatenate the vectors up to together from right to left
- Repeat the above steps for every eigenvalue of . Then concatenate the matrices corresponding to each eigenvalue together.
This is the so-called Jordan basis, the change-of-basis matrix we need to calculate the Jordan form. The order of matrices does not matter since Jordan normal form is only unique up to a permutation of Jordan blocks.
- We need to calculate the inverse of , usually by Gaussian ellimination.
- We calculate the Jordan form by .