Example of a proper node?

Question

I know that a proper node is a node with repeated eigenvalues, but with 2 linearly independent eigenvectors. An improper node has repeated eigenvalues, and 1 linearly independent eigenvector. I cannot understand how it is possible to have a proper node since the eigenvectors are determined from the eigenvalues, and if they are repeated, surely you will end up with the same eigenvectors, hence always improper nodes. Thanks in advance for the help.

Answer 1

$\begin{pmatrix}1&0\\0&1\end{pmatrix}$ has 2 linearly independent eigenvectors, but $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ has only one. And eigenvectors are not determined by eigenvalues, two matrices can have identical eigenvalues and different eigenvectors. Or even different numbers of linearly independent eigenvectors as this example shows.