A maximum modulus theorem for spectral radius and absolutely stable amplifiers

Stability problems invariably impose constraints on eigenvalues and the spectral radius, Spr
A
, of a matrix
A
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
A(z) \equiv A(z_1, z_2, \cdots , z_k)
denote a rational matrix in the
k
independent variables
z_{i}, i \rightarrow k
, which is analytic in the closed unit polydisc,
\bar{D}^k(1) \equiv \{z: |z_1| \leq 1,|z_2| \leq 1, \cdots ,|z_k| \leq 1\}
. Let
\partial\bar{D}^k(1) \equiv \{z:|z_1|=1,|z_2|=1,\cdots , |z_k|=1\}
denote the distinguished boundary of
\bar{D}^{k}(1)
. Then, Spr
A(z)\leq 1
for all
z\in|bar{D}^k(1)
if and only if Spr
A(z) \leq 1
for all
z \in \partial\bar{D}^k(1)
. 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
k
-port amplifier can always be tested by closing its
k
-ports on
k
uncoupled pure reactances. Lastly, we present an entirely new justification of the well-known fact that a reciprocal
k
-port amplifier is absolutely stable if and only if it is strictly passive.