Find a linear transformation such that $S^2 = T$ (general case) Suppose a linear transformation $T:\mathbb{R}^3\rightarrow \mathbb{R}^3$, $T\begin{pmatrix}
x\\ 
y\\ 
z
\end{pmatrix} = \begin{pmatrix}
x+y\\ 
y+z\\ 
z+x
\end{pmatrix}$ (just an example).
How do i find a linear transformation S such that $S^2=S\circ S=T$ ?
it's part of a basic linear algebra course. Would appreciate a general answer (not specific to the example above). Iv'e already tried several ways, didn't manage to get somewhere.. 
Thanks.
EDIT: The answer is supposed to be based only on basic matrices and linear transformations material, no Diagonalization and eigenvalues 

Here is an explicit counterexample:
Consider the linear transformation $T:\mathbb{R} \to \mathbb{R}$, $x \mapsto -x$. If $S:\mathbb{R} \to \mathbb{R}$, $x \mapsto ax$ (for some $a \in \mathbb{R}$) is a linear transformation satisfying $S \circ S = T$, then $-1 = T(1) = (S \circ S)(1) = a^2 \cdot 1$. But since we are working over the reals, we cannot have $a^2 = -1$. So in general, we cannot find the transformation you are looking for.
