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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

REFERENCES

(1)Archbold, J. W.Algebra (London, 1964).Google Scholar
(2)Bôcher, M.Introduction to higher algebra (New York, 1907, reprinted 1964).Google Scholar
(3)Gantmacher, F. R.The theory of matrices, vol. 1 (New York, 1959).Google Scholar
(4)Jones, J. Jr. On the Lyapunov stability criteria. J. Soc. Indust. Appl. Math. 13 (1965), 941–945.Google Scholar
(5)Rosenbrock, H. H.On linear system theory. Proc. Inst. Elec. Engrs 114 (1967), 1353–1359.Google Scholar
(6)Rosenbrock, H. H.Generalised resultant. Electronics Letters, 4 (1968), 250–251.Google Scholar
(7)Rosenbrock, H. H.Relatively prime polynomial matrices. Electronics Letters, 4 (1968), 227–228.Google Scholar
(8)Roth, W. E.The equations AX − YB = C and AX − XB = C in matrices. Proc. Amer. Math. Soc. 3 (1952), 392–396.Google Scholar
(9)Smith, H. J. S.On systems of linear indeterminate equations and congruences. Philos. Trans. Roy. Soc. London, Ser. A, 151 (1861), 293–326.Google Scholar