Do we know $A$ if we know $z^T A z$ for all $z$?

Question

Let $A$ be a matrix and suppose that

$$z^T Az$$

is known for all $z \in \mathbb{R}^d$. Is $A$ uniquely determined by all these values?

I needed this in the case that $A$ is symmetric, which I managed to prove. But I'm wondering about the general case. I suspect this to be false. Maybe a $2\times 2$-counterexample already exists.

Answer 1

No. Consider what happens if you transpose:

\begin{equation*} z^T A z = z^T A^T z \end{equation*}