I've successfully determined the eigenvalues of the transformation matrix (T = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 5 & -10 \\ 1 & 0 & 2 & 0 \\ 1 & 0 & 0 & 3 \end{pmatrix}). Now I'm wondering about the next logical step in this process. Could someone guide me on how to find (if possible) the bases for the corresponding eigenspaces? I'd greatly appreciate any insights or steps to take from this point forward. Here's my attempt:
$$\left( T-\lambda_{1} I\right) \mathbf{v} =0$$ where $\lambda_{1} = 1$, so we have:
$$\begin{pmatrix}1-1&0&0&0\\ 0&1-1&5&-10\\ 1&0&2-1&0\\ 1&0&0&3-1\end{pmatrix} \left( \begin{gathered}v_{1}\\ v_{2}\\ v_{3}\\ v_{4}\end{gathered} \right) =0$$
Thank you in advance!
$A = \begin{bmatrix}1&0&0&0\\0&1&5&-10\\1&0&2&0\\1&0&0&3\end{bmatrix}$
$\det(A-\lambda I) = \lambda^4-7\lambda^3+17\lambda^2-17\lambda+6=(\lambda-3)(\lambda-2)(\lambda-1)^2$ so we have eigenvalues $1, 1, 2, 3$
The eigenspace for $\lambda_2=2$ will simply be the nullspace of $A-\lambda_2 I$. To find this, look at the row reduction of $A-\lambda_2 I$
$\text{rref}\left(\begin{bmatrix}-1&0&0&0\\0&-1&5&-10\\1&0&0&0\\1&0&0&1\end{bmatrix}\right) = \begin{bmatrix}1&0&0&0\\0&1&-5&0\\0&0&0&1\\0&0&0&0\end{bmatrix}$
Interpreting this, this says that if $v = \begin{bmatrix}v_1\\v_2\\v_3\\v_4\end{bmatrix}$ were a vector such that $(A - \lambda_2)v = 0$ (which would mean that $Av = \lambda_2 v$) that it must be such that $\begin{cases}v_1=0\\v_2-5v_3=0\\v_4=0\\0=0\end{cases}$
This means, if we were to parametrize this by letting $v_3$ be some scalar... let's call it $t$, that we have $v_1=0, v_2=5t, v_3=t, v_4=0$, in other words that $\begin{bmatrix}0\\5\\1\\0\end{bmatrix}$ works as a basis for the eigenspace for $\lambda = 2$.
The fact that it just so happens that this was one of the columns of the original matrix is a fluke and you should not expect this to happen again in other problems.
This process is, again as pointed out in the comments, no different than what you should have learned and already been performing when being introduced to nullspaces in the previous chapter(s) before ever being shown the word "eigenspace."