The Inverse of a Nearly Banded Matrix

TL;DR

This paper presents a quantitative Nullity Theorem relating singular values of submatrices, extends it to infinite-dimensional function spaces, and discusses its implications in linear algebra and functional analysis.

Contribution

It introduces a linear relation between singular values of submatrices and extends the theorem to infinite-dimensional spaces, broadening its theoretical scope.

Findings

Establishes a linear relation between singular values of submatrices.

Extends the Nullity Theorem to function spaces.

Provides a discussion on the infinite-dimensional case.

Abstract

A quantitative form of the Nullity Theorem is presented, which establishes a linear relation between the singular values of the two submatrices involved in the theorem up to the first order. The theorem is then extended to function spaces and a corresponding form in infinite dimension is discussed.