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

Shaun Fallat; Charles R. Johnson; Alan D. Sokal

doi:10.1016/j.laa.2017.01.013

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

Shaun Fallat, Charles R. Johnson, Alan D. Sokal

Research output: Contribution to journal › Article › peer-review

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≥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.

Original language	English (US)
Pages (from-to)	242-259
Number of pages	18
Journal	Linear Algebra and Its Applications
Volume	520
DOIs	https://doi.org/10.1016/j.laa.2017.01.013
State	Published - May 1 2017

Keywords

Hadamard critical exponent
Hadamard power
Hadamard product
Hankel matrix
Stieltjes moment problem
Totally nonnegative matrix
Totally positive matrix

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2017.01.013

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title = "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≥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.",

keywords = "Hadamard critical exponent, Hadamard power, Hadamard product, Hankel matrix, Stieltjes moment problem, Totally nonnegative matrix, Totally positive matrix",

author = "Shaun Fallat and Johnson, {Charles R.} and Sokal, {Alan D.}",

note = "Publisher Copyright: {\textcopyright} 2017 Elsevier Inc.",

year = "2017",

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doi = "10.1016/j.laa.2017.01.013",

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journal = "Linear Algebra and Its Applications",

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TY - JOUR

T1 - Total positivity of sums, Hadamard products and Hadamard powers

T2 - Results and counterexamples

AU - Fallat, Shaun

AU - Johnson, Charles R.

AU - Sokal, Alan D.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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≥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.

AB - 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≥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.

KW - Hadamard critical exponent

KW - Hadamard power

KW - Hadamard product

KW - Hankel matrix

KW - Stieltjes moment problem

KW - Totally nonnegative matrix

KW - Totally positive matrix

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U2 - 10.1016/j.laa.2017.01.013

DO - 10.1016/j.laa.2017.01.013

M3 - Article

AN - SCOPUS:85010187569

SN - 0024-3795

VL - 520

SP - 242

EP - 259

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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