SPECTRAL RADIUS, KRONECKER PRODUCTS AND STATIONARITY

Abstract. We provide a stochastic proof of the inequality ρ(A⊗A+B⊗B) ≥ρ(A⊗A), where ρ(M) denotes the spectral radius of any square matrix M, i.e. max{|eigenvalues| of M}, and M⊗N denotes the Kronecker product of any two matrices M and N. The inequality is then used to show that stationarity of the bilinear model

will imply stationarity of the linear part, i.e. the linear ARMA model

for r= 1 and q= 1. Furthermore, it is shown that stationarity of the subdiagonal model, i.e. the bilinear model with b_ij=0 for i< j, again implies stationarity of its linear part, provided that the stationarity condition given by Bhaskara Rao and his colleagues is met. Interestingly, the conclusion that stationarity of the subdiagonal models, implies that the linear component models cannot be extended to the general non-subdiagonal bilinear models. The last observation is demonstrated via a simple example with p=m= 1, r= 0 and q= 2.