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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
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##### References:
 [1] N. K. Bose: Applied Multidimensional Systems Theory. Van Nostrand Reinhold C., New York, 1982. · Zbl 0574.93031 [2] W. Rudin: Function Theory in the Unit Ball of $$C^n$$. Springer, Berlin, 1980. · Zbl 0495.32001 [3] S. G. Krantz: Function Theory of Several Complex Variables. John Wiley and Sons, New York, 1982. · Zbl 0471.32008 [4] A. Fettweis G. Linnenberg: A Class of Two- Dimensional Reactance Functions With Applications. Archiv. Für Elektronik und Übertragungstechnik, vol. 34 (1980), pp. 276-278. [5] J. Gregor: On Quadratic Hurwitz Forms. Aplikace Matematiky 26 (1981), pp. 142-153. · Zbl 0457.15016 [6] T. Koga: Synthesis of Finite Passive n-Ports with Prescribed Two-variable Reactance Matrices. JEEE vol. CT- 13 (1966), pp. 31 - 52. [7] F. M. Reza: Product of Inductive and Capacitive Operators. IEE Proc. vol. 129, Pt. G. No. 5, October 1982, pp. 241-244.
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