Real degeneracy loci of matrices and phase retrieval

Let ${\mathcal A} = \{A_{1},\dots,A_{r}\}$ be a collection of linear operators on ${\mathbb R}^m$. The degeneracy locus of ${\mathcal A}$ is defined as the set of points $x \in {\mathbb P}^{m-1}$ for which rank$([A_1 x \ \dots \ A_{r} x]) \\ \leq m-1$. Motivated by results in phase retrieval we study degeneracy loci of four linear operators on ${\mathbb R}^3$ and prove that the degeneracy locus consists of 6 real points obtained by intersecting four real lines if and only if the collection of matrices lies in the linear span of four fixed rank one operators. We also relate such {\em quadrilateral configurations} to the singularity locus of the corresponding Cayley cubic symmetroid. More generally, we show that if $A_i , i = 1, \dots, m + 1$ are in the linear span of $m + 1$ fixed rank-one matrices, the degeneracy locus determines a {\em generalized Desargues configuration} which corresponds to a Sylvester spectrahedron.