In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$. Let $G \in \mathscr{M}_n(\mathbb{C})$ be the matrix defined by $$G = Y\, {}^t Z$$ where $Y, Z \in \mathscr{M}_{n,1}(\mathbb{R})$ are two column vectors such that ${}^t Y Y = {}^t Z Z = 1$. (a) Show that $G$ has rank 1 and give its image. (b) Show that $0$ and ${}^t Z Y$ are the only eigenvalues of $G$. (c) Deduce that $G \in \mathbb{M}_n(u)$. (d) Determine $\varphi_G$ when ${}^t Z Y \neq 0$. (e) Deduce that if ${}^t Z Y \neq 0$ then $$u(G) = U(0) I_n + \frac{U({}^t Z Y) - U(0)}{{}^t Z Y} G.$$ (f) Determine a simple expression for $u(G)$ when ${}^t Z Y = 0$.
In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $G \in \mathscr{M}_n(\mathbb{C})$ be the matrix defined by
$$G = Y\, {}^t Z$$
where $Y, Z \in \mathscr{M}_{n,1}(\mathbb{R})$ are two column vectors such that ${}^t Y Y = {}^t Z Z = 1$.\\
(a) Show that $G$ has rank 1 and give its image.\\
(b) Show that $0$ and ${}^t Z Y$ are the only eigenvalues of $G$.\\
(c) Deduce that $G \in \mathbb{M}_n(u)$.\\
(d) Determine $\varphi_G$ when ${}^t Z Y \neq 0$.\\
(e) Deduce that if ${}^t Z Y \neq 0$ then
$$u(G) = U(0) I_n + \frac{U({}^t Z Y) - U(0)}{{}^t Z Y} G.$$
(f) Determine a simple expression for $u(G)$ when ${}^t Z Y = 0$.