A multiplication theorem for two-variable positive real matrices. (English) Zbl 0579.94018

Consider the transformation \(s_ 1=\sqrt{(p_ 1p_ 2)}\), \(s_ 2=\sqrt{(p_ 1/p_ 2)}\) and \(p_ 1=s_ 1s_ 2\), \(p_ 2=s_ 1/s_ 2\) where \(s_ i\) and \(p_ i\), \(i=1,2\), are complex variables. The author establishes the following theorem. Let \(Z_ 1(s_ 1)\) and \(Z_ 2(s_ 2)\) be two positive real functions in \(s_ 1\) and \(s_ 2\) respectively. Then the functions \(Z_ 1(\sqrt{(p_ 1p_ 2)})\cdot Z_ 2(\sqrt{(p_ 1/p_ 2)})\) and \(Z_ 1(\sqrt{(p_ 1p_ 2)})/Z_ 2(\sqrt{(p_ 1/p_ 2)})\) are positive real in \(\{p_ 1,p_ 2\}\). A similar theorem for two-variable positive real matrices is also presented.
Reviewer: D.P.Brown


94C05 Analytic circuit theory
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