Projector matrices and properties How do I construct matrices $P\in\mathbb{C}^{\ n\ x\ n}$ such that:


*

*They are not projectors (projector: $A=A^2$)

*$Col(P)\cap Ker(P) = \{0\}$

*$\forall x \in \mathbb{C}^n,\ x = y+z\quad \left(y\in Col(P),\ z\in Ker(P)\right)$

 A: What you need is that your $P$ has no nilpotent part. That is, its Jordan form should be diagonal. In terms of blocks, your $P$ will be of the form $SXS^{-1}$ with $S$ invertible and 
$$
X=\begin{bmatrix}Y&0\\0&0\end{bmatrix},
$$
with $Y\in\mathbb C^{m\times m}$, $m\leq n$. 
At its simplest form, for instance, here is an example of such $P$:
$$
P=\begin{bmatrix}1&0&0\\ 0&2&0\\0&0&0\end{bmatrix}. 
$$
The column space is the span of the first two vectors in the canonical basis, while the kernel is the span of the third vector in the canonical basis. 
A: It was easier than I thought. 
For example $T= \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ has $T^2= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, so $T\neq T^2$.
The $col(T)=\mathbb{C}^2$, and $ker(T)=0$. Then properties 2 and 3 are satisfied.
Another example $S= \begin{bmatrix}2 & 0 \\ 0 & 0\end{bmatrix}$ has $S^2= \begin{bmatrix}4 & 0 \\ 0 & 0\end{bmatrix}$, so $S\neq S^2$.
The $col(S)=\begin{bmatrix}1\\ 0 \end{bmatrix}$, and $ker(T)=\begin{bmatrix} 0 \\ 1\end{bmatrix}$; so properties 2 and 3 are also satisfied.
