TITLE: Finding a invariant subspaces for a specific matrix
QUESTION [10 upvotes]: Good morning,
How does one find the subspaces that are invariant under $A$ for
$$A = \begin{pmatrix} 1 & 0 & 0 & 0\\  0 &2  & 0 & 0\\  0 & 0 & 2 & 0\\  0 & 0 & 0 & 3 \end{pmatrix}\  \in M_{4} (\mathbb{R}).$$
Thank you.

REPLY [3 votes]: Define $\mathscr{T}:\mathbb{R}^{4}\rightarrow\mathbb{R}^{4}$ by $\mathscr{T}(\mathbf{x})=A\mathbf{x}$ for $\mathbf{x}\in \mathbb{R}^{4},$ where $$A=\begin{pmatrix}
 1&  0&  0&  0\\ 
 0&  2&  0&  0\\ 
 0&   0&  2&  0\\ 
 0&  0&   0&  3  
\end{pmatrix}\in M_{4} (\mathbb{R}).$$
$$\mathbf{e}_{1}=(1,0,0,0)^{T},\mathbf{e}_{2}=(0,1,0,0)^{T},\mathbf{e}_{3}=(0,0,1,0)^{T},\mathbf{e}_{4}=(0,0,0,1)^{T}.$$
The complete list of $\mathscr{T}$-invariant subspaces :

Zero-dimensional subspace: $\{\mathbf{0}\}.$
One-dimensional subspaces:$$\text{Span}\lbrace \mathbf{e}_{1}\rbrace,\text{Span}\lbrace \mathbf{e}_{2}\rbrace,\text{Span}\lbrace \mathbf{e}_{3}\rbrace,
\text{Span}\lbrace \mathbf{e}_{4}\rbrace,\text{Span}\lbrace \mathbf{e}_{2}+k\mathbf{e}_{3}\rbrace(k\ne 0).$$
Two-dimensional subspaces:
$$\text{Span}\lbrace \mathbf{e}_{1},\mathbf{e}_{2}\rbrace,\text{Span}\lbrace \mathbf{e}_{1},\mathbf{e}_{3}\rbrace,\text{Span}\lbrace\mathbf{e}_{1},\mathbf{e}_{4}\rbrace,
\text{Span}\lbrace\mathbf{e}_{2},\mathbf{e}_{3}\rbrace,\text{Span}\lbrace \mathbf{e}_{2},\mathbf{e}_{4}\rbrace,\text{Span}\lbrace\mathbf{e}_{3},\mathbf{e}_{4}\rbrace,\text{Span}\lbrace\mathbf{e}_{1},\mathbf{e}_{2}+k\mathbf{e}_{3}\rbrace(k\ne 0),\text{Span}\lbrace\mathbf{e}_{4},\mathbf{e}_{2}+k\mathbf{e}_{3}\rbrace(k\ne 0).
$$
Three-dimensional subspaces:

$$\text{Span}\lbrace \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\rbrace,\text{Span}\lbrace \mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\rbrace,\text{Span}\lbrace \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{4}\rbrace,\text{Span}\lbrace \mathbf{e}_{1},\mathbf{e}_{3},\mathbf{e}_{4}\rbrace,\text{Span}\lbrace \mathbf{e}_{1},\mathbf{e}_{4},\mathbf{e}_{2}+k\mathbf{e}_{3}\rbrace(k\ne0).$$

Four-dimensional subspace:$\text{Span}\lbrace\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\rbrace.$