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Transactions of the American Mathematical Society

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Preserving positivity for rank-constrained matrices

Authors: Dominique Guillot, Apoorva Khare and Bala Rajaratnam
Journal: Trans. Amer. Math. Soc. 369 (2017), 6105-6145
MSC (2010): Primary 15B48; Secondary 26E05, 26A48
Published electronically: March 29, 2017
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Abstract: Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg [Duke Math. J. 9, 1942] and Rudin [Duke Math. J. 26, 1959]. Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension $ n$. Such characterizations for a fixed value of $ n$ are difficult to obtain, and in fact are only known in the $ 2 \times 2$ case. In this paper, using a novel and intuitive approach, we study entrywise functions preserving positivity on distinguished submanifolds inside the cone obtained by imposing rank constraints. These rank constraints are prevalent in applications, and provide a natural way to relax the elusive original problem of preserving positivity in a fixed dimension. In our main result, we characterize entrywise functions mapping $ n \times n$ positive semidefinite matrices of rank at most $ l$ into positive semidefinite matrices of rank at most $ k$ for $ 1 \leq l \leq n$ and $ 1 \leq k < n$. We also demonstrate how an important necessary condition for preserving positivity by Horn and Loewner [Trans. Amer. Math. Soc. 136, 1969] can be significantly generalized by adding rank constraints. Finally, our techniques allow us to obtain an elementary proof of the classical characterization of functions preserving positivity in all dimensions obtained by Schoenberg and Rudin.

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Additional Information

Dominique Guillot
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716

Apoorva Khare
Affiliation: Departments of Mathematics and Statistics, Stanford University, Stanford, California 94305

Bala Rajaratnam
Affiliation: Department of Statistics, University of California, Davis, California 95616

Keywords: Entrywise positive maps, rank preserving maps, rank constraint, absolutely monotonic functions, positive semidefiniteness, Loewner ordering
Received by editor(s): June 11, 2015
Received by editor(s) in revised form: September 11, 2015
Published electronically: March 29, 2017
Additional Notes: The authors were partially supported by the US Air Force Office of Scientific Research grant award FA9550-13-1-0043, US National Science Foundation grants DMS-0906392, DMS-CMG 1025465, AGS-1003823, DMS-1106642, DMS-CAREER-1352656, Defense Advanced Research Projects Agency DARPA YFA N66001-11-1-4131, the UPS Foundation, SMC-DBNKY, and an NSERC postdoctoral fellowship
Article copyright: © Copyright 2017 American Mathematical Society

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