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Mahler's matrices

Published online by Cambridge University Press:  09 April 2009

D. H. Lehmer
D. H. Lehmer
Affiliation:
The University of California, Berkeley.
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Recently K. Mahler [1] introduced a set of φ(2n) matrices of n rows and columns which form under multiplication the abelian group of the residue classes prime to 2n modulo 2n. These remarkable matrices whose elements 0, 1 and −1, have latent roots and determinants which can be given explicitly. Thus we have new examples of matrices with given elements whose powers, roots, inverses and determinants can be written down precisely. Such matrices are often useful in testing the efficacy of methods for finding these functions for a general matrix.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 4 , December 1960 , pp. 385 - 395
Copyright
Copyright © Australian Mathematical Society 1960

References

[1]Mahler, K., A Matrix Representation of the Primitive Residue Classes Modulo 2n, Proc Amer. Math. Soc. 8 (1957), 525–531.Google Scholar
[2]Titchmarsh, E. C., The Theory of the Riemann Zeta-function, Oxford (1951), p. 10.Google Scholar
[3]Vandiver, H. S. and Nicol, C. A.suggest that this statement be known as the Dedekind-Hölder theorem. [see Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 917–918].Google Scholar