Published online by Cambridge University Press: 20 January 2009
The study of the primitive solutions of the equation
where A = (aij) is an n × n matrix whose elements are rational integers, was begun a long time ago. In most cases this equation occurred incidentally in another theory; for instance Jordan encountered it in connection with linear differential equations having algebraic solutions, Minkowski in connection with quadratic forms and Turnbull in geometry. An important fact about these matrices is that any unimodular matrix can be represented as the product of matrices with finite period.
1. It will be understood throughout this note that every matrix considered, unless otherwise stated, has all its elements rational integers. Further we shall understand by integer rational integer unless some other meaning is given explicitly.
Among the papers dealing with the problem under consideration are
Baker, H. F., Proc. London Math. Soc. (1), 35 (1903), 379–384.Google Scholar
Jordan, C., Journ. f. Math. 84 (1878), 89–215, especially 112.Google Scholar
Minkowski, H., Journ. f. Math. 101 (1887), 196–202; also in Gesammelte Abhandlungen 1 (Leipzig-Berlin, 1911), 212–218. (Cf. A. Speiser, Theorie der Gruppen von endlicher Ordnung ((3), Berlin, 1937), §67).Google Scholar
A more complete bibliography is available in the reports of: Wedderburun, J. H. M., Lectures on Matrices (American Math. Soc. Coll. Publications, 17, New York, 1934);Google Scholarvan der Waerden, B. L., Gruppen ron linearen Transformationen (Berlin, 1935); C. C.CrossRefGoogle ScholarMacDuffee, , The Theory of Matrices (Berlin, 1933).CrossRefGoogle Scholar
3. It is interesting to observe that it is always possible to find a form of degree n in n variables which is invariant under the transformation determined by an assigned n × n matrix A with | A | = ± 1. To see this denote the latent roots of A by ρ1,......,ρn and denote by a non-trivial solution of the system of equations
Then the form
in the n variables u 1, ......, un is easily shown to be invariant under the transformation determined by A.
4. Here and elsewhere we find it convenient to use the summation convention.
6. When we say that A has period r we understand that Ar = 1 and that As ≠ 1 for any s = 1, 2, ......, r − 1.
7. See e.g. Turnbull, H. W. and Aitken, A. C.The Theory of Canonical Matrices (London-Glasgow, 1932), Ch. 5.Google Scholar
9. See the papers of Vaidyanathaswamy referred to in 1. The case r=n corresponds to the cyclic permutation of n elements; in this case we have to use the well-known relation
10. See the book of Albert referred to in 5, Ch. 4, § 5.
11. Similarly it can be seen that an n x n matrix of period r, whose characteristic polynomial is the power of an irreducible polynomial, must be the direct sum of irreducible matrices each of which has its characteristic function of degree φ(r) Such a matrix can always be obtained as the regular representation of a root of unity contained in an algebraic number field of degree n. CfTaussky, Olga and Todd, John, Proc. Royal Irish Academy, 46A (1940), 1–11.Google Scholar
13. See e.g. the book of Turnbull and Aitken referred to in 7, Ch. 1, §6.