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arXiv · 2606.18173

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

Adam Humeniuk·Gabriel Jarry-Bolduc·Patrick Pascua·Nejaunie Williams
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Abstract

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.

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BibTeXRIS

Adam Humeniuk, Gabriel Jarry-Bolduc, Patrick Pascua, Nejaunie Williams. 2026-06-16. An algorithm to exactly compute minimal upper bounds in the Loewner order. https://arxiv.org/abs/2606.18173

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