 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.