Regular polynomial matrices having relatively prime determinants
- 1 May 1969
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 65 (3) , 585-590
- https://doi.org/10.1017/s0305004100003364
Abstract
It is shown that a necessary and sufficient condition that two regular polynomial matrices T, U have relatively prime determinants is that the equation TX + YU = E, where E is a constant matrix, has a unique solution with the degrees of X, Y less than the degrees of U, T respectively. This is a generalization of a well-known theorem for scalar polynomials. An alternative form for the condition in terms of the non-vanishing of a determinant, corresponding to the resultant of two scalar polynomials, is also obtained together with an equivalent determinant of lower order.This publication has 4 references indexed in Scilit:
- On linear system theoryProceedings of the Institution of Electrical Engineers, 1967
- On the Lyapunov Stability CriteriaJournal of the Society for Industrial and Applied Mathematics, 1965
- The Equations AX - YB = C and AX - XB = C in MatricesProceedings of the American Mathematical Society, 1952
- XV. On systems of linear indeterminate equations and congruencesPhilosophical Transactions of the Royal Society of London, 1861