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Nonnegative rank depends on the field

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Abstract

We present an example of a subfield \(\mathcal {F}\subset \mathbb {R}\) and a matrix A whose conventional and nonnegative ranks equal five, but the nonnegative rank with respect to \(\mathcal {F}\) equals six. In other words, A can be represented as a sum of five rank-one matrices with nonnegative real entries but not as a sum of five rank-one matrices with nonnegative entries in \(\mathcal {F}\).

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Acknowledgements

I am grateful to Kaie Kubjas for an interesting discussion on the topic and helpful suggestions on the presentation of the result. A part of this discussion held in 2015 in Aalto University in Helsinki, and I am grateful to Kaie for inviting me and to the colleagues from the university for their hospitality. I would like to thank Till Miltzow for a discussion and for pointing me to [14]. I am grateful to the Editor-in-Chief and Associate Editor who were handling my submission for very detailed specific comments concerning its content and to anonymous reviewers for careful reading, valuable comments, and kind words of encouragement.

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  1. Moscow Institute of Physics and Technology, 9 Institutskiy pereulok, 141701, Dolgoprudnyy, Russia

    Yaroslav Shitov

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  1. Yaroslav Shitov
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Correspondence to Yaroslav Shitov.

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Shitov, Y. Nonnegative rank depends on the field. Math. Program. 186, 479–486 (2021). https://doi.org/10.1007/s10107-019-01448-2

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