Sharp uniform lower bounds for the Schur product theorem

10/08/2019

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by   Apoorva Khare, et al.

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By a result of Schur [J. Reine Angew. Math. 1911], the entrywise product M ∘ N of two positive semidefinite matrices M,N is again positive. Vybiral (2019) improved on this by showing the uniform lower bound M ∘M≥ E_n / n for all n × n real or complex correlation matrices M, where E_n is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 1999] and to positive definite functions. Vybiral then asked if one can obtain similar uniform lower bounds for higher entrywise powers of M, or when N ≠ M, M. In this short note, we affirmatively answer both of these questions. In addition, our lower bounds - which we show are tracial Cauchy-Schwartz inequalities - are sharp.