Projectivization of a linear vector field
= Projectivization of a linear vector field
A complex-linear endomorphism $A$ of $\mathbb C^{n+1}$ induces a <holomorphic vector field> on <Complex projective space> by differentiating the projective transformations $[z]\mapsto[e^{tA}z]$. Its zeroes are precisely the projective eigenlines of $A$; a diagonalizable $A$ with distinct eigenvalues therefore gives exactly $n+1$ zeroes.