An algorithm to exactly compute minimal upper bounds in the Loewner order

The Loewner order on Hermitian matrices is a partial order that compares matrices in terms of positive semidefiniteness. The Loewner order plays a key role in many fields such as optimization, numerical linear algebra, control theory, operator theory, and quantum information. A fundamental difficulty is that two or more Hermitian matrices do not necessarily have a unique minimal upper bound (or maximal lower bound). In this paper, we propose an iterative method to exactly compute a minimal upper bound for any finite collection of $n\times n$ Hermitian matrices. It is shown that the algorithm terminates in at most $n$ iterations. The exactitude of the algorithm is proved using standard results from finite-dimensional linear algebra. A self-contained proof of an algebraic characterization of minimality originally explored by Stott is provided. We illustrate the algorithm in examples and also provide an implementation of the algorithm in Python.

Source: arXiv:2606.18173v1 - http://arxiv.org/abs/2606.18173v1 PDF: https://arxiv.org/pdf/2606.18173v1 Original Link: http://arxiv.org/abs/2606.18173v1