Let \(k\) be a field and \(A \in \Mat_{n\times n}(k)\), and suppose that the characteristic polynomial factors as \(\chi_A(x) = \pm\prod_{i=1}^r (x-\lambda_i)^{m_i}\) with \(\lambda_1, \ldots, \lambda_r \in k\) distinct. For each \(i\), put \(N_i \coloneqq A - \lambda_i I\). The following procedure produces \(P \in \GL_n(k)\) such that \(\inverseof{P} A P = \JCF(A)\) is a Jordan canonical form of \(A\).
Let \(\ell_i\) be the least integer \(\ell \geq 1\) with \(\rank N_i^{\ell} = \rank N_i^{\ell+1}\). Then \[ n = \rank N_i^0 > \rank N_i > \cdots > \rank N_i^{\ell_i} = \rank N_i^{\ell_i+1} = \rank N_i^{\ell_i+2} = \cdots, \] \(\dim_k \ker N_i^{\ell_i} = m_i\), and \(\ell_i\) is the size of the largest Jordan block with eigenvalue \(\lambda_i\).
The eigenspace \(E_{\lambda_i} = \ker N_i\) has dimension \(n - \rank N_i\), the number of Jordan blocks with eigenvalue \(\lambda_i\). For \(0 \leq j \leq \ell_i\), put \(K_{i,j} \coloneqq \ker N_i \cap \im N_i^{j}\), so that \[ \ker N_i = K_{i,0} \supseteq K_{i,1} \supseteq \cdots \supseteq K_{i,\ell_i} = 0 \quad\text{and}\quad \dim_k K_{i,j} = \rank N_i^{j} - \rank N_i^{j+1}. \]
Choose a basis of \(K_{i,\ell_i - 1}\) and extend it successively to bases of \(K_{i,\ell_i-2}, \ldots, K_{i,0}\). Each basis vector \(v\) chosen at the stage of \(K_{i,s-1}\) is an eigenvector lying in \(\im N_i^{s-1}\).
For each such \(v\), solve \(N_i^{s-1} w = v\), for instance by row reducing the augmented matrix \([\,N_i^{s-1} \mid v\,]\); the system is consistent because \(v \in \im N_i^{s-1}\). The Jordan chain of \(v\) is \(N_i^{s-1}w = v,\ N_i^{s-2}w,\ \ldots,\ N_i w,\ w\).
Let \(P\) be the matrix whose columns are the Jordan chains of all the vectors chosen in step (3), for all \(i\), each chain listed consecutively in the order of step (4). A chain of length \(s\) for \(\lambda_i\) contributes a Jordan block of size \(s\) with eigenvalue \(\lambda_i\) to \(\inverseof{P} A P\).