Sufficient Condition for a matrix to be null

Question

Let $A=[a_{ij}]$ be a matrix of $M_n(\mathbb{C})$. Suppose that for any permutation $\sigma$ of the set $\{1,\ldots,n\}$ and any complex numbers $u_1,\ldots,u_n \in \mathbb{C}$ of modulus one (i.e. $|u_k|=1$) we have $$ \sum_{k=1}^{n} u_ka_{k,\ \sigma(k)}=0. $$ How can we show that $A=0$?

Answer 1

The above condition says that for any vector $\vec{u} = (u_1,\ldots,u_n)$ whose entries each have unit modulus, $A \vec{u} = 0$. As there are $n$ $\mathbb C$-linearly independent vectors on the sphere of radius $\sqrt{n}$ in $\mathbb C^n$, this implies that there are $n$ linearly independent vectors in the nullspace of $A$. Said differently, the nullspace of $A$ is all of $\mathbb{C}^n$, so $A$ must be the zero matrix.