Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity

Abstract

Entrywise powers of symmetric matrices preserving positivity, monotonicity or convexity with respect to the Loewner ordering arise in various applications, and have received much attention recently in the literature. Following FitzGerald and Horn [J. Math. Anal. Appl., 1977], it is well-known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness. Similar phase transition phenomena have also recently been shown by Hiai (2009) to occur for monotonicity and convexity. In this paper, we complete the characterization of all the entrywise powers below and above the critical exponents that are positive, monotone, or convex on the cone of positive semidefinite matrices. We then extend the original problem by fully classifying the positive, monotone, or convex powers in a more general setting where additional rank constraints are imposed on the matrices. We also classify the entrywise powers that are super/sub-additive with respect to the Loewner ordering. Finally, we extend all the previous characterizations to matrices with negative entries. Our analysis consequently allows us to answer a question raised by Bhatia and Elsner (2007) regarding the smallest dimension for which even extensions of the power functions do not preserve Loewner positivity.Date: October 8, 2014. 2010 Mathematics Subject Classification. 15B48 (primary); 15A45, 26A48, 26A51 (secondary).

Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

“…Power Symmetric Matrices and Construction of PPT States. This section has been motivated by the work on "Power symmetric matrics" by Sinkhorn [81] and its generalization by Bapat, Jain and Prasad [5] on one hand and preservation of positivity of a block matrix under taking powers of blocks, the so-called Schur or Hadamard product by Choudhury [16] and Guillot, Khare and Rajaratnam [34] on the other. The purpose is to illustrate the interesting interplay rather than the utmost generality.…”

“…In that spirit, Guillot, Khare and Rajaratanam [34] give interesting further developments, but reveal that actions of replacing α by other numbers or relaxing the conditions limit the possibility of preserving the positive semi-definiteness and that of its application to the construction of PPT states.…”

“…Jain and K. Manjunatha Prasad [5]. Techniques developed by D. Choudhury [16] and D. Guillot, A. Khare and B. Rajaratnam [34] help to some extent. The fourth section is devoted to the concept of unitary equivalence of a matrix to its transpose and its general form obtained by S.R.…”

Hadamard powers of some positive matrices

“…Motivated by problems occurring in statistics, Guilllot, Khare and Rajaratnam have been studying various problems related to Hadamard powers. See [7,8].…”

A Panorama of Positivity. I: Dimension Free

“…Rank constraints and other Loewner properties. Another approach to generalize Theorem 6.1 is to examine other properties of entrywise functions such as monotonicity, convexity, and super-additivity (with respect to the Loewner semidefinite ordering)[78,68]. Given a set V ⊂ P N (I), recall that a function f :I → R is • positive on V withrespect to the Loewner ordering if f [A] ≥ 0 for all 0 ≤ A ∈ V ; • monotone on V with respect to the Loewner ordering if f [A] ≥ f [B] for all A, B ∈ V such that A ≥ B ≥ 0; • convex on V with respect to the Loewner ordering if f [λA + (1 − λ)B] ≤ λf [A] + (1 − λ)f [B] for all λ[0, 1] and all A, B ∈ V such that A ≥ B ≥ 0; • super-additive on V with respect to the Loewner ordering if f [A +…”