Georgia Institute of Technology College of Sciences

How close is a random matrix to being singular? This question measures the stability of random linear systems. Classical models treat every direction equally, but in many settings different directions have different scales or variances. We study what happens when a random matrix is deformed by a fixed invertible matrix. Even a diagonal deformation raises a basic question: is stability controlled by the weakest direction, or by the collective geometry of all directions? I will explain why standard estimates miss this geometry and how random subspaces lead to sharper bounds.