Spectral properties of a sequence of matrices connected to each other via Schur complement and arising in a compartmental model

Abstract

We consider a sequence of real matrices An which is characterized by the rule that An−1 is the Schur complement in An of the (1,1) entry of An, namely −en, where en is a positive real number. This sequence is closely related to linear compartmental ordinary differential equations. We study the spectrum of An. In particular,we show that An has a unique positive eigenvalue λn and {λn} is a decreasing convergent sequence. We also study the stability of An for small n using the Routh-Hurwitz criterion.

Keywords

• schur complement
• routh-hurwitz criterion
• elementary symmetric polynomials
• linear compartmental model
• latency phase