A system of determinant equations: solutions, or at least existence thereof? While doing some physics, I have encountered an interesting problem, which reduces to the following:
Given a collection of scalars $a_i,b_j$ and real, symmetric $N\times N$ matrices $M_i$, we wish to find scalars $x_i$ that simultaneously solve the $N$ determinant equations
$$
0 = \det\left[ \sum_{i=0}^N (a_i + b_j x_i)M_i\right]
$$
for $j=1,\ldots,N$. Under what conditions does a solution exist, and when it does, how does one find it in general?
I have explicitly solved such a problem in a specific $N=2$ case, which wasn't very hard, but I am for now stumped as to how one would tackle it at the level of generality I present here.
In the cases I'm interested in, the elements of $M_i$ have a rather systematic form (I can provide the lengthy-ish definition if needed), but the resulting matrices are not mutually commuting, so no simplifications can be made based on simultaneous diagonalization. I suppose the problem would be quite a lot easier if that was the case.
 A: Not a solution, but a geometric interpretation of your problem that might be useful to think about:
For a given $x = (x_0, \dots x_N)$, let $Q_x = \sum_i x_iM_i$. In particular, $Q_a = \sum_i a_iM_i$.
The collection of all $N\times N$ symmetric matrices forms an $\binom{N+1}{2}$-dimensional vector space and the $M_i$, assuming they are linearly independent, span an $N+1$ dimensional subspace $\cal Q$ of it. The $Q_x$ for various $x$ are exactly the elements of this space. For any fixed $x$, the set $L_x = \{Q_a + tQ_x\mid t\in \Bbb R\}$ is an affine line in $\cal Q$. The $b_j$ form $N$ points on that line.
$\det Q = 0$ forms an $N$ dimensional hypersurface in $\cal Q$. I do not know enough to say much about what this hypersurface looks like. But "hypersurface" itself is likely not entirely accurate. It will be a variety, which allows for the surface to bifurcate in certain locations (for example, two crossing lines counts as a variety, but not a curve, because the point of intersection does not look like $\Bbb R$).
Picking $x$ to solve the equation amounts to choosing a line passing through $Q_a$ that intersects the hypersurface in $N$ points whose distances from $Q_a$ are in exactly the ratios of the $b_j$. That is, if the $j$-th point of intersection is at a distance of $d_j$ from $Q_a$, then for all $j, \frac{d_j}{d_1} = \frac{b_j}{b_1}$. (Getting $d_1$ to be the right distance from $Q_a$ to satisfy the $b_1$ equation can be accomplished by simply rescaling $x$.)
It seems to me that this would be very hard to accomplish as $N$ gets bigger. So tight limitations on $M_i$ and $b_j$ will be necessary for solutions.
