Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

TL;DR

This paper compiles open research questions in matrix computations, focusing on linear systems, eigenvalues, low-rank approximation, and randomized sketching, arising from a collaborative workshop at the Simons Institute.

Contribution

It presents a curated set of open problems in numerical linear algebra developed through interdisciplinary discussions at a specialized workshop.

Findings

Identifies key open questions in iterative solvers and eigenvalue problems.

Highlights challenges in low-rank approximation and randomized sketching.

Bridges theoretical computer science and numerical analysis in problem formulation.

Abstract

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was organized at the Simons Institute for the Theory of Computing program on Complexity and Linear Algebra in Fall 2025. The complexity and numerical solution of linear algebra problems is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions.