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On the Canonical Form of a Rational Integral Function of a Matrix

Published online by Cambridge University Press:  20 January 2009

D. E. Rutherford
D. E. Rutherford
Affiliation:
St Andrews University.
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It is well known that the square matrix, of rank n−k + 1,

which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 3 , Issue 2 , July 1932 , pp. 135 - 143
Copyright
Copyright © Edinburgh Mathematical Society 1932

References

page 135 note 1 Proc. Akad. Amsterdam, 35 (1932), 157.Google Scholar

page 135 note 2 References to the original investigation by Frobenius, and to others, are given by Turnball and Aitken, Canonical Matrices (Glasgow, 1932), 81.Google Scholar

page 139 note 1 See Canonical Matrices, 67.Google Scholar