A generalization of Cayley's theorem

Abstract

CAYLEY (MACDUFFEE, C.C.: Theory of matrices. New York: Chelsea, 1946, p. 50) had proved in 1856 that if A is a non-singular matrix, with elements in a field K (denoted A~K), and if PAQ=A, and P+E and Q+E are nonsingular, where P~K, Q~K and E is the identity matrix; then there exists a matrix M satisfying A _+ M are non-singular and such that P = (A + M) (A M ) 1 and Q = (A + M)-1 ( A M). Also that the converse holds. The object of the present investigation is to show how to deal with case in which A is an arbitrary matrix. SIEGEL [Equivalence of quadratic forms Amer. J. Math. 63, 658-680. (1941)] has given an application of CAYLEY'S theorem. This generalization (Th. 2) has an application in the corresponding theory over algebraic number fields. The author hopes to publish it elsewhere. In w 1, we develop some terminology and prove some lemmas, that are essentially due to SIEGEL [l~ber die analytische Theorie der quadratischen Formen III. Ann. Math. 38, 212-291 (1937)]. The treatment given here is, however, much simpler for the results needed here. In w 2, the main results are proved. The proofs are on classical lines, the important step being the statement of correct results.