Given that $x^TAx = 0$, can I call $A$ a null transformation? I am wondering what to call the matrix $A$ given that $x^TAx=0$ for all $x$, which implies that $A=\textbf{0}$. Would I then simply call $A$ the zero matrix or maybe does it have a special name?
 A: No, it does not imply that $A = \mathbf 0$. Try any skew-symmetric matrix!! Indeed, working over $\Bbb R$, we can prove that this hypothesis implies that $A$ must be skew-symmetric.
Since $x^\top Ax = 0$ for all $x$, we use the standard "polarization" trick, substituting $x+y$ for $x$:
\begin{align*}
0=(x+y)^\top A(x+y) &= x^\top Ax + x^\top Ay + y^\top Ax + y^\top Ay \\ &= x^\top Ay + y^\top Ax = x^\top (A+A^\top)y.
\end{align*}
But if $x^\top By = 0$ for all $x$ and $y$, we must have $B=\mathbf 0$ (take the standard basis vectors, if you want). So we conclude that $A^\top = -A$.
(The fundamental formula here at the last step is that, identifying a scalar and a $1\times 1$ matrix, we have
$$x^\top Ay = (x^\top Ay)^\top = y^\top A^\top x = y^\top A^\top x.)$$
A: It is not true what you write. In fact, your relation holds if, and only if, the symmetric part of $A$ vanishes, i.e., if $A$ is skew-symmetric.
To see this, just compute the $i$-th derivatives. For every $x$ you have
$$
0=\partial_i(Ax\cdot x)= A e_i\cdot x+ Ax\cdot e_i=(A+A^T)x\cdot e_i \, .
$$
Just set $x=e_j$ to get that, under your assumption
$$
(A+A^T)e_i\cdot e_j=0
$$
for every $i,j$.
So to answer your question, the right name to give to those matrices is skew-symmetric.
