QII.A
Invariant lines and eigenvalues and vectors
Invariant subspaces and stable subspace analysis
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In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$.
The goal here is to show that a subspace $F$ of $E$ is stable by $f$ if and only if $F = \bigoplus_{i=1}^{p} (F \cap E_i)$.
II.A.1) Show that every subspace $F$ of $E$ such that $F = \bigoplus_{i=1}^{p} (F \cap E_i)$ is stable by $f$.
II.A.2) Let $F$ be a subspace of $E$ stable by $f$ and $x$ a non-zero vector of $F$.
Justify the existence and uniqueness of $(x_i)_{1 \leqslant i \leqslant p}$ in $E_1 \times \cdots \times E_p$ such that $x = \sum_{i=1}^{p} x_i$.
II.A.3) If we denote $H_x = \{i \in \llbracket 1, p \rrbracket \mid x_i \neq 0\}$, $H_x$ is non-empty and, up to reordering the eigenvalues (and the eigenspaces), we can assume that $H_x = \llbracket 1, r \rrbracket$ with $1 \leqslant r \leqslant p$. Thus we have $x = \sum_{i=1}^{r} x_i$ with $x_i \in E_i \setminus \{0\}$ for all $i$ in $\llbracket 1, r \rrbracket$.
We denote $V_x = \operatorname{Vect}(x_1, \ldots, x_r)$.
Show that $\mathcal{B}_x = (x_1, \ldots, x_r)$ is a basis of $V_x$.
II.A.4) Show that for all $j$ in $\llbracket 1, r \rrbracket$, $f^{j-1}(x)$ belongs to $V_x$ and give the matrix of the family $(f^{j-1}(x))_{1 \leqslant j \leqslant r}$ in the basis $\mathcal{B}_x$.
II.A.5) Show that $(f^{j-1}(x))_{1 \leqslant j \leqslant r}$ is a basis of $V_x$.
II.A.6) Deduce that for all $i$ in $\llbracket 1, p \rrbracket$, $x_i$ belongs to $F$ and conclude.