On Hadamard powers of positive semi-definite matrices
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- by Jnaneshwar Baslingker and Biltu Dan
- Proc. Amer. Math. Soc. 151 (2023), 1395-1401
- DOI: https://doi.org/10.1090/proc/16187
- Published electronically: January 13, 2023
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Abstract:
Consider the set of scalars $\alpha$ for which the $\alpha$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots , n-3\}\cup [n-2, \infty )$. A natural question is “what is the possible form of the set of such $\alpha$ for a fixed p.s.d. matrix with non-negative entries?”. In all examples appearing in the literature, the set turns out to be union of a finite set and a semi-infinite interval. In this article, examples of matrices are given for which the set consists of a finite set and more than one disjoint interval of positive length. In fact, it is proved that the number of such disjoint intervals can be made arbitrarily large, by giving explicit examples of matrices.
The case when the entries of the matrices are not necessarily non-negative is also considered.
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Bibliographic Information
- Jnaneshwar Baslingker
- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore-560012, India
- ORCID: 0000-0001-6365-1263
- Email: jnaneshwarb@iisc.ac.in
- Biltu Dan
- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore-560012, India
- MR Author ID: 1351126
- Email: biltudan@iisc.ac.in
- Received by editor(s): January 19, 2022
- Received by editor(s) in revised form: June 9, 2022
- Published electronically: January 13, 2023
- Communicated by: Javad Mashreghi
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1395-1401
- MSC (2020): Primary 15B48, 15A45
- DOI: https://doi.org/10.1090/proc/16187
- MathSciNet review: 4550337