Matrix intersection problems for conditioning
Volume 112 / 2017
Banach Center Publications 112 (2017), 195-210
MSC: 15A12, 65F35.
DOI: 10.4064/bc112-0-11
Abstract
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.
