Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop
It addresses foundational problems in linear algebra for researchers in computational mathematics and computer science, but is incremental as it synthesizes existing discussions rather than presenting new solutions.
This document compiles open questions in matrix computations, such as iterative solvers and eigenvalue problems, arising from a workshop that brought together researchers from theoretical computer science and numerical analysis to identify key challenges in the field.
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.