A maximum modulus theorem for spectral radius and absolutely stable amplifiers

Stability problems invariably impose constraints on eigenvalues and the spectral radius, Spr , of a matrix emerges as an important concept. Unfortunately, the spectral radius of a matrix does not qualify as a norm. Nevertheless, with the aid of the Lyapunov lemma we prove the following: let denote a rational matrix in the independent variables , which is analytic in the closed unit polydisc, . Let denote the distinguished boundary of . Then, Spr for all if and only if Spr for all . In addition to pointing out several obvious generalizations, we also employ the above theorem to give a rigorous proof of the long accepted conjecture that the absolute stability of a -port amplifier can always be tested by closing its -ports on uncoupled pure reactances. Lastly, we present an entirely new justification of the well-known fact that a reciprocal -port amplifier is absolutely stable if and only if it is strictly passive.