Groups of infinite matrices
Waldemar Hołubowski · Cambridge University Press eBooks · 2007
We show that in the group of infinite invertible column-finite matrices over an associative ring R , every element is a product of a row- and column-finite matrix and a unitriangular matrix. Moreover we prove that its subgroup of banded matrices is generated by strings (block-diagonal matrices, with finite blocks along the main diagonal). Introduction Let R be an associative ring with 1. Let GL c (∞, R ) denote the group of infinite ℕ × ℕ column-finite matrices over R , and GL rc (∞, R ) its subgroup of row- and column-finite matrices. A systematic study of normal subgroups of GL c (∞, R ) in the case of division rings was initiated by A. Rosenberg. Research continued in works of Maxwell, Robertson, Arrell, Arrell and Robertson, Hausen, Thomas and others. We refer to for a comprehensive survey. In this paper we are interested in results concerning generators of subgroups and a special form of an element in the case of an arbitrary associative ring of coefficients. We prove: Theorem 1.1 Every element of GL c (∞, R ) is a product of an invertible row- and column-finite matrix and an upper unitriangular matrix. The matrix a ∈ GL c (∞, R ) is called n -banded if a ij = 0 for all i, j such that | j – i | > n and either a i + n, i ≠ 0 or a i,i + n ≠ 0 for at least one index i .