Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

Abstract

We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t \ge r-2. We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices.

arXiv commentLaTex2e, 18 pages; Version 2 incorporates Corrigendum giving a corrected proof of Theorem 3.2

References in corpus (2)

• Structured matrices, continued fractions, and root localization of polynomials
• Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

Cited by in corpus (13)

• A Generalization of Linear Positive Systems with Applications to Nonlinear Systems: Invariant Sets and the Poincaré-Bendixson Property
• Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples
• Variation diminishing linear time-invariant systems
• On the sign patterns of entrywise positivity preservers in fixed dimension
• Totally positive kernels, Polya frequency functions, and their transforms
• The Poisson binomial distribution -- Old & New
• Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values
• Balanced truncation of $k$-positive systems
• Parametrizations of $k$-Nonnegative Matrices: Cluster Algebras and $k$-Positivity Tests
• Multiply positive functions, critical exponent phenomena, and the Jain-Karlin-Schoenberg kernel
• Random Fibonacci Words via Clone Schur Functions
• Total-positivity preservers
• On matrices whose exponential is a P-matrix