## Konveks inverteerbare keëls en die Cohen-Lewkowicz interpolasiestelling

##### Abstract

Convex cones, de ned on any algebra with a unit element, which are closed under inversion (convex invertible cones or cics for short) are studied. We establish that the intersection of cics is a cic. For a set X contained in a unital algebra, we denote by C (X) the cic generated by X. To get C (X) from its generating set X, we provide an inductive construction. We set X0 = R+ X; and Xk+1 is obtained from Xk by taking convex combinations of members in Xk and their inverses. The union of this increasing sequence Xk produces C (X). One of our main goals is to study the Lyapunov inequality in association with matrix cics. We restrict ourselves to the real Lyapunov inequality, both nonstrict and strict, and to cics of real matrices. We start by studying the structure of real matrix cics, along with other related properties such as subcics, nonsingularity and similarity. Matrix inertia plays a major role in characterizing nonsigular cics. Next we look at the various connections real matrix cics have with the nonstrict and strict, real Lyapunov inequality. For a given real matrix A, we study the two sets of all possible solutions to the nonstrict and strict, real Lyapunov inequality, for this matrix A. Under certain conditions on A, we are able to determine all the possible solutions to the strict, real Lyapunov inequality, for this matrix A. We are particularly interested in the event where for two given real matrices A and B, it follows that all the solutions to the nonstrict, real Lyapunov inequality for A, are also solutions to the nonstrict, real Lyapunov inequality for B. This describes the Lyapunov order between two real matrices A and B. Among other things, we study the main properties of the Lyapunov order and introduce the cic of all real matrices satisfying the Lyapunov order with a xed real matrix. We also study various cics in the algebra of real, rational functions R. Thus R consists of all functions f that can be expressed as the ratio of two real polynomials in a complex variable, where inversion is given by involution f ! 1f . These cics include the cic of positive, real rational functions (PR) and the cic of positive, real, odd rational functions (PRO). We make use of control theory to study the cic PRO. We end the study by showing that if a real matrix B belongs to the cic generated by a real matrix A, there is at least one f 2 PRO such that B = f(A). To nd such an f 2 PRO for two given real matrices A and B is a variation on Nevanlinna-Pick interpolation introduced by N. Cohen and I. Lewkowicz, which we shall refer to as the Cohen-Lewkowicz interpolation problem. Hereby we have established a connection between the matrix cics and the function cics we have studied.