Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T13:35:04.245Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular greatest common divisor of two polynomial matrices

Published online by Cambridge University Press:  24 October 2008

S. Barnett
S. Barnett
Affiliation:
School of Mathematics, University of Bradford, Yorkshire
Get access Rights & Permissions [Opens in a new window]

Abstract

Let T(λ) and V(λ) be two polynomial matrices having dimensions l x l and m x l respectively, with T(λ) regular and of degree n and V(λ) of degree at most n – 1. It has recently been shown that a necessary and sufficient condition for T and V to be relatively right prime is that a certain nlm x nl matrix R(T, V) have full rank. It is shown here that if T and V have a greatest common right divisor D(λ), then provided D is regular, its degree k is equal to n – (1/l) rank R. Furthermore, if R˜. denotes the matrix of the first (n – k) lm rows of R, then it is shown that the last (n – k) l columns of R˜ are linearly independent and that the coefficient matrices of D can be obtained by expressing the remaining columns of R˜ in terms of this basis.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 2 , September 1972 , pp. 161 - 165
Copyright
Copyright © Cambridge Philosophical Society 1972

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Barnett, S.Greatest common divisor of two polynomials. Linear Algebra and Appl. 3 (1970), 7–9.CrossRefGoogle Scholar
(2)Barnett, S.Greatest common divisor of several polynomials. Proc. Cambridge Philos. Soc. 70 (1971), 263–268.CrossRefGoogle Scholar
(3)Barnett, S.Matrices in control theory (London, 1971).Google Scholar
(4)Macduffee, C. C.The theory of matrices, p. 35 (New York, 1956).Google Scholar
(5)Rosenbrock, H. H.State-space and multivariable theory (London, 1970).Google Scholar
(6)Rowe, A.The generalised resultant matrix, University of Manchester Institute of Science arid Technology, Control Systems Centre Report No. 156 (1971). To appear J. Inst. Maths. Appl. (1972).Google Scholar