Let $$T(\begin{pmatrix}a&b\\ c&d\end{pmatrix})=\begin{pmatrix}a&a+b\\ b+c&c+d\end{pmatrix}$$
Find All eigenvectors (not only the Linear independence) of $T$ over $\mathbb{Z}_5$
My attempted :
First I found Matrix Representation $[T]_B$ when $B$ is the standard basis :
$$[T]_B=\begin{pmatrix}1&0&0&0\\ 1&1&0&0\\ 0&1&1&0\\ 0&0&1&1\end{pmatrix}$$
$[T]_B$ is lower triangular matrix then the characteristic polynomial is $P(\alpha) = (\alpha-1)^4$
and the Eigenvalue are $\alpha_{1,2,3,4} = 1$
and to find characteristic vectors we need to find the solution space of $[T]_B - I =0$
$\begin{pmatrix}0&0&0&0\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{pmatrix} \rightarrow \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\end{pmatrix}$
the general solution :
$s\begin{pmatrix}1\\ 0\\ 0\\ 0\end{pmatrix} + t\begin{pmatrix}0\\ 1\\ 0\\ 0\end{pmatrix} + w\begin{pmatrix}0\\ 0\\ 1\\ 0\end{pmatrix}$
and the characteristic vectors are :
$$\left\{\begin{pmatrix}1\\ \:0\\ \:0\\ \:0\end{pmatrix},\begin{pmatrix}0\\ \:1\\ \:0\\ \:0\end{pmatrix},\begin{pmatrix}0\\ \:0\\ \:1\\ \:0\end{pmatrix}\right\}$$
Is this correct and what the question mean by finding all the characteristic vector (not only the Linear independence) ?
thanks