$J$-expanding mtrix functions and their role in the analytical theory of electrical circuits

A. V. Efimov ; V. P. Potapov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 28 (1973) no. 1, pp. 69-140

Résumé

Chapter I establishes the essential properties of the $\mathscr A$-matrix of a passive multipole depending on the number of its branches. These properties are based on Langevin's theorem. A classification of the basic objects of investigation:$J$-expanding matrix-functions (class $\mathfrak M$), and also positive matrix functions (class $\mathfrak B$ ), is introduced. Chapter II gives an account of a theory of matrix functions of class $\mathfrak M$. It also investigates the simplest (elementary and primary) matrices of this class. The fact is established that elementary (and primary) factors can be split off from a given matrix of class $\mathfrak M$. In particular, the factorizability of a rational reactive matrix of class $\mathfrak M$ is established. Chapters III–IV set forth a theory of various subclasses of matrix functions of class $\mathfrak M$: $\mathfrak M_{sl}$, $\mathfrak M_{cgl}$, $\mathfrak M_{lr}$. The realizability of the matrix functions of each of these subclasses as $\mathscr A$-matrices of passive multipoles with the corresponding provision for branches is established. The fact that they are realizable is proved by the construction of a corresponding multipole. The last chapter is concerned with a generalization of Darlington's theorem, which leads to a realization of functions of the subclasses $\mathfrak M_{clr}$ and $\mathfrak M_{cglr}$ as $\mathscr A$-matrices or $z$-matrices of dissipative multipoles.