A maximum modulus theorem for spectral radius and absolutely stable amplifiers
Dante C. Youla · IEEE Transactions on Circuits and Systems · 1980
Stability problems invariably impose constraints on eigenvalues and the spectral radius, SprA, of a matrixAemerges 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: letA(z) \equiv A(z_1, z_2, \cdots , z_k)denote a rational matrix in thekindependent variablesz_{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, SprA(z)\leq 1for allz\in|bar{D}^k(1)if and only if SprA(z) \leq 1for allz \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 ak-port amplifier can always be tested by closing itsk-ports onkuncoupled pure reactances. Lastly, we present an entirely new justification of the well-known fact that a reciprocalk-port amplifier is absolutely stable if and only if it is strictly passive.