Baslingker, J and Dan, B (2023) ON HADAMARD POWERS OF POSITIVE SEMI DEFINITE MATRICES. In: Proceedings of the American Mathematical Society, 151 (4). pp. 13951401.

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Abstract
Consider the set of scalars α for which the αth Hadamard power of any n × n positive semidefinite (p.s.d.) matrix with nonnegative entries is p.s.d. It is known that this set is of the form {0, 1, . . ., n − 3} ∪ [n − 2, ∞). A natural question is “what is the possible form of the set of such α for a fixed p.s.d. matrix with nonnegative entries?”. In all examples appearing in the literature, the set turns out to be union of a finite set and a semiinfinite 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 nonnegative is also considered.
Item Type:  Journal Article 

Publication:  Proceedings of the American Mathematical Society 
Publisher:  American Mathematical Society 
Additional Information:  The copyright for this article belongs to the Authors. 
Keywords:  Hadamard power; Positive semidefinite 
Department/Centre:  Division of Physical & Mathematical Sciences > Mathematics 
Date Deposited:  20 Mar 2023 07:26 
Last Modified:  30 Mar 2023 06:59 
URI:  https://eprints.iisc.ac.in/id/eprint/81012 
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