Let $\Omega$ be an open set in $\mathbb{C}^n$, $f:\Omega \rightarrow \mathbb{C}$ be a $C^2$ function on $\Omega$. Let $H_f (a)$ be the complex Hessian of $f$ at $a\in \Omega$. In Rudin's book Function Theory in Unit Ball $\mathbb{C}^n$, it is claimmed that
$\langle H_f(a)b,b\rangle=0$ for all $a\in \Omega$ and for all $b\in \mathbb{C}^n$ if and only if each entry of $ H_f(a)$ is zero.
Now my question is how the condition $\langle H_f(a)b,b\rangle=0$ can imply that each entry of $ H_f(a)$ is zero.
Please help. Thank you.