On clans of non-negative matrices

Abstract

A clan is a compact connected topological semigroup with identity. Professor A. D. Wallace has raised the following question [9]: Is a clan of real n X n matrices with non-negative entries, which contains the identity matrix, necessarily acyclic? That is to say, do all of the Alexander-Cech cohomology groups with arbitrary coefficients (in positive dimensions) vanish? In this paper the slightly stronger result, that any non-negative matrix clan is contractible, is obtained. This follows from the result, interesting in itself, that a compact group of non-negative matrices is finite (Theorem 2). The author wishes here to express his sincere gratitude to Professor R. J. Koch for his helpful advice and encouragement. The set of order n non-negative matrices is denoted by Nn. The real and complex general linear groups of order n are represented by Gl(n, R) and Gl(n, C), respectively. The semigroup terminology used is that of [8]; in particular, K denotes the minimal ideal of a clan 8, E denotes the set of idempotents of S, and for eCE, H(e) is the maximal subgroup of S containing e. An iseomorphism is an isomorphism which is also a homeomorphism. The topology of Nn is any locally convex topology; for example, the topology of Euclidean n2space. The equation M= diag(A, B) means that M is the matrix which, in 2 X 2 block form, has the square submatrix A in the upper left corner, the square submatrix B in the lower right corner, and zero entries elsewhere. The kXk identity matrix is denoted by Ik when used as a submatrix. The set of eigenvalues of a matrix M is denoted by S(M). The well-known theorem [1, p. 80] that a non-negative matrix M has a real eigenvalue r such that if X C S(M), then I X ? < r is used without proof. Also used without proof is the following theorem, due to Karpelevich [3], and stated in less than full generality: