Inverses of matrices and matrix-transformations

Abstract

The first author wishes to acknowledge with gratitude many instructive consultations with J. A. Schatz, and correspondence with M. S. MacPhail. Let A = (ank), n, k = 1, 2, , be a matrix of complex numbers. Let D be the set (linear sequence space) of sequences x = { x, } such that y =Ax is defined; y being the sequence {yn }I, where Yn = EkankXk for each n. Let R be the set of all Ax, xeD. We call D and R the domain and range of A. They are linear subspaces of (s), the space of all sequences. To emphasize the distinction between inverse matrix and inverse transformation, we denote Ax by T(x), thus defining T:D>R, and investigate, under various hypotheses: (a) the existence of right, left, and two-sided inverses for A, denoted by A', 'A, A-', (b) the same for T, denoted by T', 'T, T-1, (c) connections between (a) and (b). By A' we mean any matrix satisfying AA' = I, the identity matrix. By we mean any function T':R->D satisfying T(T'(x)) =x for all xCR. The other symbols are interpreted similarly. By T' we mean there exists at least one T'. Similarly for the others. Our main results concern row-finite matrices, i.e. such that almost all the elements in each row are zero; column-finite matrices, i.e. matrices whose transpose is row-finite; and reversible matrices, i.e. matrices A such that for each convergent sequence y, the equation y =Ax has a unique solution (we shall see that if A is row-finite, reversibility is equivalent to the existence of a unique solution for all y). A discussion is given of the constants Cn of Banach [1, p. 50 ] which appear in the inverse transformation of a reversible matrix. Let E be the (countably infinite-dimensional) set of sequences x such that xn=0 for almost all n, (c) the set of convergent sequences. Clearly, DDE for all A; A is row-finite if and only if D = (s), column-finite if and only if AxCE whenever xCE, reversible if and only if RD(c), and T is 1-1 (i.e. to each y R corresponds exactly one xCD; A is 1-1 will mean that the associated T is 1-1).