We know that we are able to put a 2x2 matrix $A$ into the following Jordan normal form: $A=PJP^{-1}$ with
$J = \begin{pmatrix} \lambda_1 & a \\ 0 & \lambda_2 \end{pmatrix}$
Where $a=0$ or $a=1$.
Of course, if $\lambda_1\neq \lambda_2$, then our matrix $A$ has two distinct eigenvalues, and thus has corresponding linearly independent eigenvectors. Thus putting into Jordan normal form corresponds simply to diagonalisation with $a=0$. The columns of $P$ are the LI eigenvectors of $A$.
The same is possible if $\lambda_1=\lambda_2=\lambda$, but $\lambda$ has geometric multiplicity 2. We may also set the columns of $P$ to be LI vectors corresponding to the $\lambda$ eigenvalue, and we are dagonalising $A$.
If we have $\lambda_1=\lambda_2=\lambda$ with geometric multiplicity $1$, then we set $a=1$. If we write $P=(c_1,c_2)$ as the columns then $c_1$ is an eigenvector of $A$, and $c_2$ satisfies $(A-\lambda I)c_2 = c_1$. That is, $c_1$ is in the image of $(A-\lambda I)$: the nullspace of this matrix $(A-\lambda I)$ is a basis for the image space.
I am trying to wrap my head around why it makes sense for this to be so, and would appreciate any input!