This paper presents the non-linear generalization of a previous work on matrix differential models [1]. It focusses on the construction of approximate solutions of first-order matrix differential equations Y ′ (x) = f (x,Y (x)) using matrix-cubic splines. An estimation of the approximation error, an algorithm for its implementation and illustrative examples for Sylvester and Riccati matrix diff...

Sylvester’s conjecture states that, given n distinct noncollinear points in a plane, there exists a connecting line of two of the points such that no other point is incident with the line. First a proof is given of the six-point Sylvester conjecture from a constructive axiomatization of plane incidence geometry. Next ordering principles are studied that are needed for the seven-point case. This...

The resultant is an algebraic expression, computable in a finite number of arithmetic operations from the coefficients of two univariate polynomials, that vanishes if, and only if, the two polynomials have common zeros. The paper considers formal resultant for degree-deficient polynomials (polynomials whose actual degree is lower than their assumed degree). Some key properties of the resultant ...

A new method for the design of reduced-order observers for descriptor systems with unknown inputs is presented. The approach is based on the generalized constrained Sylvester equation. Sufficient conditions for the existence of the observer are given.

We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and ⋆-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of 2× 2 block matrix representations of the equations a...

In this work, we investigate the interval generalized Sylvester matrix equation AXB+ CXD = F and develop some techniques for obtaining outer estimations for the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator which causes reducing computational complexity to cubic, compared to Kronecker product form. We then propose an iterati...