total nonnegativity
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2019 ◽  
Vol 578 ◽  
pp. 446-461
Author(s):  
Dominique Guillot ◽  
Jiaru Wu
Keyword(s):  

2018 ◽  
Vol 61 (4) ◽  
pp. 836-847
Author(s):  
Kevin Purbhoo

AbstractWe consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point V of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if V is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix A preserves stability of polynomials if and only if A is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya–Schur theory of Borcea and Brändén.


2018 ◽  
Vol 467 (1) ◽  
pp. 148-170
Author(s):  
Mohammad Adm ◽  
Jürgen Garloff ◽  
Mikhail Tyaglov

2016 ◽  
Vol 508 ◽  
pp. 214-224
Author(s):  
Mohammad Adm ◽  
Jürgen Garloff
Keyword(s):  

Author(s):  
Shaun M. Fallat ◽  
Charles R. Johnson

Totally nonnegative matrices arise in a remarkable variety of mathematical applications. This book is a comprehensive and self-contained study of the essential theory of totally nonnegative matrices, defined by the nonnegativity of all subdeterminants. It explores methodological background, historical highlights of key ideas, and specialized topics. The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references.


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