Complex vector space identity Let $A$ be an $n \times n$ complex matrix and let $u \in \mathbb{C}$ be a column vector. Suppose $u^*Au=0$ for all $u$.  What's the most straightforward way to see that $A=0$?

If one takes for $u$ the vector $\begin{pmatrix}x_1&\ldots& x_n \end{pmatrix}$ then the proposed equation becomes a quadratic form that is zero for all the values of the indeterminates $x_1, \ldots,x_n$ so its coefficients must  all be zero.
