## Linear transformations with nontrivial stable subspaces are realized by block diagonal matrices

Let $V$ be an $n$-dimensional vector space over a field $F$ and let $\varphi : V \rightarrow V$ be a linear transformation. Suppose $W \subseteq V$ is a subspace of dimension $m$ which is stable under $\varphi$. That is, $\varphi[W] \subseteq W$.

1. Prove that there is a basis $B$ for $V$ such that $M^B_B(\varphi) = \left[ \begin{array}{c|c} A & B \\ \hline 0 & C \end{array} \right]$, where $A$ has dimension $m \times m$ and $C$ has dimension $(n-m) \times (n-m)$.
2. Prove that if there is a subspace $W_2$, also invariant under $\varphi$, such that $V = W \oplus W_2$, then there is a basis $E$ for $V$ such that $M^E_E(\varphi) = \left[ \begin{array}{c|c} A & 0 \\ \hline 0 & C \end{array} \right]$, where $A$ has dimension $m \times m$ and $C$ has dimension $(n-m) \times (n-m)$.
3. Prove conversely that if there is a basis $T$ of $V$ such that $M^T_T(\varphi)$ is block diagonal as in part (2), then there exist $\varphi$-invariant subspaces $W_1$ and $W_2$ such that $V = W_1 \oplus W_2$.

We discussed block matrices in more depth in this previous exercise.

1. Let $B_1 = \{w_1, \ldots, w_m \} \subseteq W$ be an ordered basis. Since $B_1 \subseteq V$ is linearly independent, we may extend $B_1$ to a basis $B = \{w_1, \ldots, w_m, v_1, \ldots, v_{n-m} \}$ of $V$. Since $\varphi[W] \subseteq W$, if we write $\varphi(w_i)$ using this basis, we have $\varphi(w_i) = \sum \alpha_i w_i + \sum \beta_i v_i$ and the $\beta_i$ are all 0. Thus we have $M^B_B(\varphi) = \left[ \begin{array}{c|c} A & B \\ \hline 0 & C \end{array} \right]$.
2. Let $B_1 = \{ w_1, \ldots, w_m \} \subseteq W$ and $B_2 = \{v_1, \ldots, v_{n-m}\} \subseteq W_2$ be ordered bases. We claim that $E = \{w_1, \ldots, w_m, v_1, \ldots, v_{n-m}\}$ is a basis for $V$. Certainly it is a generating set. Now suppose $\sum \alpha_i w_i + \sum \beta_i v_i = 0$. Since $V$ is the direct sum of $W$ and $W_2$, we have $\sum \alpha_i w_i = 0$ and $\sum \beta_i v_i = 0$. Thus $\alpha_i = 0$ and $\beta_i = 0$, and so $E$ is linearly independent. Note that, as in part (1), with respect to this basis we have $\varphi(w_i) = \sum \alpha_i w_i + \sum \beta_i v_i$, and $\beta_i = 0$. Likewise $\varphi(v_i) = \sum \alpha_i w_i + \sum \beta_i v_i$ with $\alpha_i = 0$. Thus we have $M^E_E(\varphi) = \left[ \begin{array}{c|c} A & 0 \\ \hline 0 & C \end{array} \right]$.
3. Let $T = \{t_i\} \subseteq V$ be an ordered basis such that $M^T_T(\varphi)$ is block diagonal; say $M^T_T(\varphi) = \left[ \begin{array}{c|c} A & 0 \\ \hline 0 & C \end{array} \right]$, where $A$ has dimension $m \times m$ and $C$ has dimension $(n-m) \times (n-m)$. Let $T_1 = \{ t_1, \ldots, t_m\}$ and $T_2 = \{t_{m+1}, \ldots, t_n\}$. Let $W_1 = \mathsf{span}\ T_1$ and $W_2 = \mathsf{span}\ T_2$. Certainly $W_1$ and $W_2$ are both stable under $\varphi$, and $V = W_1 \oplus W_2$.