Eigenspace decomposition of a linear involution
= Eigenspace decomposition of a linear involution
{title2=$P_\pm=(I\pm F)/2$}
Let $F$ be a <linear map> on a <vector space> over a field of characteristic different from two, with $F^2=I$. Then
$$
V=\ker(F-I)\oplus\ker(F+I),\qquad
P_\pm=\tfrac12(I\pm F).
$$
The projections satisfy $P_+P_-=0$ and $P_++P_-=I$, giving existence and uniqueness of the decomposition. In characteristic two, the two signs coincide and the conclusion need not hold.