Vector Space and Matrix Methods in Signal and System Theory

TL;DR

This paper explores how linear algebra, abstract algebra, and functional analysis tools can be applied to signal processing and system theory, focusing on operator equations and their diverse interpretations.

Contribution

It provides a broad study of operator equations in various vector space contexts, highlighting their relevance to signal and system analysis.

Findings

Operator equations are central to signal processing methods.

Different interpretations of A x = b lead to diverse analytical approaches.

The mathematical framework supports applications in approximation, optimization, and big data.

Abstract

The tools, ideas, and insights from linear algebra, abstract algebra, and functional analysis can be extremely useful to signal processing and system theory in various areas of engineering, science, and social science including approximation, optimization, parameter identification, big data, etc. Indeed, many important ideas can be developed from the simple operator equation A x = b by considering it in a variety of ways. If x and if b are vectors from the same or, perhaps, different vector spaces and A is an operator, there are three interesting questions that can be asked which provide a setting for a broad study.