Minimum Variance Solution of Underdetermined Systems of Linear Equations

TL;DR

This paper introduces the Minimum Variance (MV) solution for underdetermined linear systems, providing a new approach to select solutions with specific desirable properties, and derives a simple closed-form expression.

Contribution

It proposes the MV solution as a novel alternative to the Minimum Norm solution for underdetermined systems, with a clear mathematical formulation and properties.

Findings

MV solution has unique properties compared to MN solution

Derived a simple closed-form expression for the MV solution

Discussed the properties and potential applications of the MV solution

Abstract

A system of linear equations is said underdetermined when there are more unknowns than equations. Such systems may have infinitely many solutions. In this case, it is important to single out solutions possessing special features. A well known example is the Minimum Norm (MN) solution, which is the solution having the least Euclidean norm. In this note, we consider another useful solution, related with the MN one, which we call the Minimum Variance (MV) solution. We discuss some of its properties and derive a simple, closed form expression.