Given an analytic space $V$ with an isolated singularity $p$, a Poincaré-Hopf type of index, $\mathrm {Ind}(X,V,p)$ is associated to every holomorphic vector field $X$ tangent to $V$ for which $p$ is an isolated zero. In this paper this topological index is related to the algebraic multiplicity $\mu _v(X,p)$. In particular, it is shown that the set of indices $\mathrm {Ind}(X,V,p)$, where $X$ is tangent to $V$ with an isolated zero at $p$, admits a minimum which is reached for $X$ in the open dense subset of vector fields of smallest $V$-multiplicity.