I am using this picture for reference. Firstly, please correct me if my intuition is wrong. For an equation Ax=b, x being in the row space (or x having some component in the row space and some in the nullspace, i.e. $x = x_r + x_n $), then Ax is in the column space - and vice versa. Also x is in the nullspace means Ax = 0 and if b is in nullspace of $A^T$, there is no solution.
Assuming that is correct, for an eigenvector x, $Ax=\lambda x$ so the x is scaled after this transformation. Is there any indication as to where x, the eigenvector, lies? Whether that is in the row space or some combination of the row space and nullspace, $x=x_r+x_n.$ And lastly, generally, the eigenvector x and the vector Ax would be in different spaces? Ax would be in the column space and x would be in the row space or outside of it, which is what I am really trying to get an understanding of.
Consider the matrix
$$M = \begin{pmatrix}1 & 0 & 1\\\ 1 & 1 & 0 \\\ 0 & 0 & 0 \end{pmatrix}$$
One of its eigenvectors is:
$$ x = \begin{pmatrix}0 \\\ 1 \\\ 0\end{pmatrix}$$
Notice that this is a column of $M$. Since it is a column of $M$, it cannot be orthogonal to all of the columns of $M$, and hence it cannot be in the left null space ($M^T x \ne 0$). But it is not in the row space of $M$ either.
$$ \begin{pmatrix} a & b & c \end{pmatrix} \begin{pmatrix}1 & 0 & 1\\\ 1 & 1 & 0 \\\ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} a+b & b & a \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \end{pmatrix}$$
This has no solution, since $b = 1$ and $a = 0$, so $a = -b$ is a contradiction.
But any eigenvector $x$ of $M$, with $Mx = \lambda x$, is of course in the column space, since its pre-image is $\lambda^{-1}x$. Since $x$ and $\lambda^{-1}x$ are proportional to each other, they must be in the same subspace.
See also How do we know that nullspace and row space of a matrix are orthogonal complements? for more on this subtle matter. We can't assume that each vector is either in the column space or in the null space (or equivalently for the row space and left null space). Some vectors are in neither. Of course, if the matrix is of full rank, then any vector will be in the column space (unless it is the zero vector).