Matrix intersection problems for conditioning

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>Conditioning of a nonsingular matrix subspace is addressed in terms of its best conditioned elements. The problem is computationally challenging. Associating with the task an intersection problem with unitary matrices leads to a more accessible approach. A resulting matrix nearness problem can be viewed to generalize the so-called Löwdin problem in quantum chemistry. For critical points in the Frobenius norm, a differential equation on the manifold of unitary matrices is derived. Another resulting matrix nearness problem allows locating points of optimality more directly, once formulated as a problem in computational algebraic geometry.</p></div>