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Regular polynomial matrices having relatively prime determinants

Published online by Cambridge University Press:  24 October 2008

S. Barnett
S. Barnett
Affiliation:
Department of Mathematics, University of Technology, Loughborough, Leicestershire
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 3 , May 1969 , pp. 585 - 590
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

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