Remark

— 1) It follows from Prop. 6 of VII, p. 34, that, if $U$ is similar to a diagonal block of Jordan matrices $(J_k)$, then the number of $J_k$ of the form $U_{m,\alpha}$ (for given $m$ and $\alpha$) is uniquely determined by $U$.

2) More generally, if $U$ is similar to a diagonal block of Jordan matrices $U_{m_i, \alpha_i}$, then the similarity invariants of $U$ can be readily calculated by a method modelled on that presented in VII, p. 25, Remark 3 : write all the $(X - \alpha_i)^{m_i}$ with the same $\alpha$ on the same line, in decreasing order of exponents, and complete with 1’s to have lines whose lengths are equal to the order of $U$; this done, the similarity invariants of $U$ are obtained, in decreasing order of indices, by forming the products of terms in the same column. For example, for the matrix

we write

and the similarity invariants are 1, 1 and $(X - 2)\ (X - 3)'$

By noticing that the minimal polynomial of the Jordan matrix $U_{m, \alpha}$ is $(X - a)''$, and that it is equal to its characteristic polynomial, we obtain the following result :