On the Associativity of Infinite Matrix Multiplication

TL;DR

This paper investigates the conditions under which the multiplication of infinite matrices is associative, focusing on the support restrictions and summability notions necessary for consistent matrix product behavior.

Contribution

It characterizes the specific conditions and summability notions required for the associativity of infinite matrix multiplication.

Findings

Identifies conditions for the well-definedness of infinite matrix products.

Provides characterizations of associativity using summability concepts.

Clarifies the structure needed for associativity in infinite matrix multiplication.

Abstract

A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the intersection of the support of each row of the first factor with the support of each column of the second factor must be finite. Multiplication is hence not completely defined, but restricted to a specific relation on infinite matrices. In order for the product of three infinite matrices $A$, $B$, and $C$ to behave in an associative manner, the middle factor, $B$, must link $A$ and $C$ in three ways: (i) $AB$ and $BC$ must both be defined; (ii) $A(BC)$ and $(AB)C$ must both be defined; and, finally, (iii) $A(BC)$ must equal $(AB)C$. In this article, these conditions are studied and are characterized in various ways by means of summability notions akin to those of formal calculus.