Exercise 10, § 5

a) Show that, over an arbitrary field $K$, the matrix

is similar to the Jordan matrix $U_{n,\lambda}$ if $\alpha \neq 0$. b) If $\lambda \neq 0$ and $K$ is of characteristic 0, show that $(U_{n,\lambda})^m$ is similar to the Jordan matrix $U_{n,\lambda^m}$ for every integer $m$ (positive or negative). c) Show that $(U_{n,0})^m = 0$ for $m \geq n$. For $0 < m < n$, let $n - 1 = km + q$ with $0 \leq q < m$; show that $(U_{n,0})^m$ has $q + 1$ similarity invariants equal to $X^{k+1}$ and $m - q - 1$ similarity invariants equal to $X^k$ (arrange the canonical basis of $K^n$ in a suitable order). d) Similarly, determine the similarity invariants of $f(U_{n,\lambda})$ for $f \in K[X]$. e) Discuss the same problems for $K$ of characteristic $p \neq 0$.