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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
MSC:
 94C05 Analytic circuit theory
Keywords:
positive real functions
Full Text:
References:
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