Suppose I have a system of $\nu$ quadratic forms, \begin{align*} x^T A_1 x &= 0 \\ x^T A_2 x &= 0 \\ &\vdots \\ x^T A_\nu x &= 0, \end{align*} where $x \in \mathbb{R}^{\mu}$ and each $A_i \in \mathbb{R}^{\mu \times \mu}$. Suppose it's also the case that $x^T x = 1$. Suppose none of the $A_i$ are proportional to the identity matrix.
If the $A_i$ are algebraically independent (e.g., orthonormal with respect to the Hilbert-Schmidt inner product), is it the case that there always exists a solution to this system if $\mu = \nu+1$? If not, are there any restrictions we can put on the $A_i$ so that there will be a solution?
I have observed this phenomenon computationally in every case I have tried so I am hoping/guessing there is some general theory here but I really have no experience in these areas of mathematics.
Related questions are given here, here, here and here, although I was hoping for a bit more detail. Some references on this material would also be helpful.