Sharp uniform lower bounds for the Schur product theorem
share
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.
READ FULL TEXT