orthogonal idempotents in matrix space Two matrix $A$, $B$ are orthogonals if the product is zero, $AB=0$ and $A$ is idempotent if $AA=A$.
Is it possible to identify all idempotent orthogonal matrix $3X3$ with elements in the complex set?
I can see that Must be satisfied the systems $AA=A$, $BB=B$ and $AB=0$, but Is there a more specific characterization about the entries, or how should $A$ and $B$ be?
 A: Concerning dimensions: Recall that the (Grassmannian) space of k-dimensional subspaces in n-dimensional complex space is a manifold of (complex) dimension $(n-k)\times k$ [As local coordinates one may take the space of $(n-k)\times k$ matrices]
Both $A$ and $B$ are projections (though not orthogonal). Each are completely specified by their image and kernel. Let us fix the ranks of $A$ and $B$, respectively.
Thus, let $0\leq d\leq 3$ be the dimension of ${\rm im} A$ (so $\dim \ker A=3-d$).
The dimension of the space of such $A$'s is $d\times (3-d) + (3-d)\times  d = 2 d (3-d)$ (the first coming from specifying ${\rm im} A$, the second from $\ker A$ (which are both arbitrary a part from having trivial intersection).
Let  $0\leq k\leq 3-d$ be the dimension of ${\rm im} B$. Here we have $k\leq 3-d$ because ${\rm im} B \subset \ker A$. Again the dimension of such $B$'s is given by specifying the image which has dimension
$k\times (3-d-k)$ (because we are restricted to stay within the kernel of $A$) and the kernel, which has dimension $(3-k)\times k$ (no restrictions).
All in all the total dimension of the manifold of such $A$ and $B$'s is given by
$$ 2 d(3-d) + k(3-d-k)+(3-k)k$$
So for example if you look at the manifold of $A$ and $B$ that are both restricted to having rank $d=k=1$ then the dimension of this space is
$ 2\times 2+ 1\times 1+ 2\times 1=7$, one less than if you did not have the constraint $AB=0$.
If you want orthogonality to be symmetric, so that $AB=BA=0$ then also ${\rm im} A\subset \ker B$ which reduces the dimension further (as $0\leq d \leq 3-k$ in that case). The resulting dimension is $2(d(3-d) + k(3-k) -kd)$
