Projection Matrices and $I - A$ my question is concerning complements of spaces and subtraction from $I$ -Identity matrix in $\mathbb R^3$
Given a projection matrix $A$, we are able to get the projection on the orthogonal compliment (null space $A^{T}$  ) by subtracting the projection matrix from I. I don't understand why a projection matrix is different from any other matrix with rank < columns n.
Given some Matrix $A$ that goes from $\mathbb{R}^3\to\mathbb{R}^3$ with a dimension of 2 and nullspace dimension of 1 such as
\begin{bmatrix}
1 & 4 & 2 \\
2 & 5 & 4 \\
3 & 6 & 6
\end{bmatrix}
why doesn't subtracting this from $I$ give me a matrix that gives combinations of the nullspace of $A^T$?
When i subtract I get a rank 3 matrix, not a rank 1 matrix I was expecting. Nor is any vectors in this in the nullspace of $A^T$
\begin{bmatrix}
0 & -4 & -2 \\
-2 & -4 & -4 \\
-3 & -6 & -5
\end{bmatrix}
What is it i'm not understanding right now?
EDIT:
I seemed to have overlooked some properties of the projection matrices. My apologies folks. PP= P and also the symmetry aspect.
Thanks for the clarification and help.
 A: The null space of any (even non-orthogonal) projection $P$ is given by all vectors $y=(I-P)x$: this can be seen by looking at $Py=P(I-P)x=Px-P^2x=Px-Px=0$.
Consider a matrix $A$. It has a pseudo-inverse $A^{\dagger}$. There is an interesting link between pseudo-inverses and orthogonal projections, that took me a while to understand at first. The Wikipedia article is here but it doesn't have much in the way of detail.
The projection matrix $P=AA^{\dagger}$ is an orthogonal projection: the "orthogonal" part means that, by definition, its range is the orthogonal complement of its nullspace. $P$ is an orthogonal projection onto the range of $A$, and $I-P$ is an orthogonal projection onto the kernel of $A^*$, or onto the orthogonal complement of the range of $A$ - also the range of $P$. $A^*$ means the conjugate transpose; it is like $A^T$, just extended to complex matrices.
Anyway, although $P$ and $I-P$ are orthogonal complements of one another, and these $P$ exist for any matrix, $A$ and $I-A$, when $A$ is arbitrarily any matrix, have no requirement on them to be orthogonally complementary, whereas the projection matrices $P$ and $I-P$ are provably orthogonally complementary to one another.
Since the trail of Wikipedia articles I read the other day are hard to parse, I will happily clarify anything.
