Kronecker products of defective matrices: Some spectral properties and their implications on observability

The eigenvector analysis of the Kronecker product of defective matrices is revisited, and is brought to bear in characterizing observability of linear systems with Kronecker-product state matrices. Specifically, a new proof is given for a classical result on the eigenspace dimension for the Kronecker product of two Jordan chains, which highlights that the dimension is essentially dependent on the off-diagonal entries' zero pattern. Based on the spectral characterization of the Kronecker product, observability conditions are then developed for linear systems with Kronecker-product state matrices. Simple examples from two application domains are also included, to illustrate how the spectral and observability results can inform stochastic-systems analysis.