zbMATH — the first resource for mathematics

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
Full Text: EuDML
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.