A new method of Jordan's normalization (Q2741589)

This short note proposes an improvement over the conventional method in finding a basis with respect to which a given finite square (real or complex) matrix \(A\) can be represented in its Jordan canonical form.NEWLINENEWLINENEWLINERecall that for such a task, we need to find, for each eigenvalue \(\lambda\) of \(A\), all order-\(k\) generalized eigenvectors \(x\) \(((A-\lambda I)^kx=0\) and \((A-\lambda I)^{k-1}x\neq 0)\) which are linearly independent with \(k\) less than or equal to the smallest integer \(\ell\) for which rank\((A-\lambda I)^\ell= \text{rank}(A-\lambda I)^{\ell+1}\). The basis can then be obtained as the union of \(\{x,(A-\lambda I)x, \dots,(A-\lambda I)^{k-1} x\}\) for different \(\lambda,x\) and \(k\).NEWLINENEWLINENEWLINEThe present author proposes: (1) find a basis for ker\((A-\lambda I)\), and (2) when its number (the geometric multiplicity of \(\lambda)\) is less than the algebraic multiplicity of \(\lambda\), for each \(y_1\) in this basis, find one solution \(y_{j+1}\) of \((A-\lambda I)y_{j+1}=y_j\) iteratively for \(j=1,2,\dots\) until no solution exists. In this case, \(\{y_j\}\) forms part of the Jordan basis. The ease of computation for this method seems quite obvious.