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.
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