Abstract
We study ranks of the \(r\text {th}\) Hadamard powers of doubly nonnegative matrices and show that the matrix \(A^{\circ r}\) is positive definite for every \(n\times n\) doubly nonnegative matrix A and for every \(r>n-2\) if and only if no column of A is a scalar multiple of any other column of A. A particular emphasis is given to the study of rank, positivity and monotonicity of Hadamard powers of rank two, positive semidefinite matrices that have all entries positive.
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Acknowledgements
The author thanks Professor Roger A. Horn and the two anonymous referees for their valuable suggestions that improved the readability of the paper. The author especially thanks one of the referees to suggest the use of Schur complements in the proof of Theorem 8, that led to much simplification of the proof. Financial support from SERB MATRICS grant number MTR/2018/000554 is also acknowledged.
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Communicated by Kenneth Berenhaut.
Dedicated To Professor Rajendra Bhatia.
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Jain, T. Hadamard powers of rank two, doubly nonnegative matrices. Adv. Oper. Theory 5, 839–849 (2020). https://doi.org/10.1007/s43036-020-00066-6
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DOI: https://doi.org/10.1007/s43036-020-00066-6