Sharp nonzero lower bounds for the Schur product theorem

Abstract

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. Vybíral (2019) improved on this by showing the uniform lower bound M • M ≥ En/n for all n × n real or complex correlation matrices M , where En is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 1999] and to positive definite functions. Vybíral then asked if one can obtain similar uniform lower bounds for higher entrywise powers of M , or when N = M, M . A natural third question is to obtain a tighter lower bound that need not vanish, over infinitedimensional Hilbert spaces as well. In this short note, we affirmatively answer all three questions by extending and refining Vybíral's result. As a special case, the above bound of En/n can be improved to En/rk(M ). In addition, our lower bounds -which we show are tracial Cauchy-Schwartz inequalities -are sharp. We end with some consequences for positive definite functions on groups, metric spaces, and Hilbert spaces. Notation:(1) A positive semidefinite matrix is a complex Hermitian matrix with non-negative eigenvalues. Denote the space of such n × n matrices by P n = P n (C). ( 2) The Loewner ordering on n × n complex matrices is the partial order such that M ≥ N if and only if M − N ∈ P n . (3) We say that a matrix in P n is a real/complex correlation matrix if it has all diagonal entries 1, and all entries real/complex respectively. (4) The Schur product of two (possibly rectangular) m × n complex matrices A = (a ij ), B = (b ij ) equals the m × n matrix A • B with (i, j) entry a ij b ij . (5) Given a fixed integer n ≥ 1, let e = e(n) := (1, . . . , 1) T ∈ C n ; and E n := ee T ∈ P n is the matrix of all ones. (6) Given a matrix M n×n and a subset J ⊂ {1, . . . , n}, let M J×J denote the principal submatrix of M corresponding to the rows and columns indexed by J.

Lower Bounds for the Error of Quadrature Formulas for Hilbert Spaces

“…The first generalization of Proposition 2 deals with matrices with reduced rank. Independently, it was also observed in [7]. Theorem 14.…”

“…The matrix V T W is formed by the scalar products of the column vectors of V and W , respectively. Using an orthogonal projection onto their common linear span (which has dimension at most 2r), we can find X, Z ∈ R 2r×r such that X T Z = V T W , which is even stronger than (7). Let us observe that Theorem 3 is obtained by considering particular orthonormal bases of H(M d ).…”

“…Open Problem 1. We conjecture that Theorem 16 holds true with 1 r instead of 1 2r in (6), see also [7,Theorem 1.9]. This would allow to improve the error bound in Corollary 18 6 Randomly chosen sample points…”