On matrices with common invariant cones with applications in neural and gene networks

Motivated by a differential continuous-time switching model for gene and neural networks, we investigate matrix theoretic problems regarding the relative location and topology of the dominant eigenvectors of words constructed multiplicatively from two matrices A and B. These problems are naturally associated with the existence of common invariant subspaces and common invariant proper cones of A and B. The commuting case and the two-dimensional case are rich and considered analytically. We also analyze and recast the problem of the existence of a common invariant polyhedral cone in a multilinear framework, as well as present necessary conditions for the existence of low dimensional common invariant cones.