Matrix positivity preservers in fixed dimension. I

Belton, Alexander C. R.; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai

doi:10.1016/j.aim.2016.04.016

Cited by 20 publications

(47 citation statements)

References 51 publications

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“…More specifically, we are seeking functions f : N → C with the property that f • K is totally nonnegative on X whenever K is. The following result from [5] shows that for general kernels, not many functions f have this property. Proof.…”

Section: Totally Nonnegative Kernelsmentioning

confidence: 99%

Total nonnegativity of GCD matrices and kernels

Guillot¹,

Wu²

2019

Linear Algebra and its Applications

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Let X = (x 1 , . . . , x n ) be a vector of distinct positive integers. The n × n matrix S = S(X) := (gcd(x i , x j )) n i,j=1 , where gcd(x i , x j ) denotes the greatest common divisor of x i and x j , is called the greatest common divisor (GCD) matrix on X. By a surprising result of Beslin and Ligh [Linear Algebra and Appl. 118], all GCD matrices are positive definite. In this paper, we completely characterize the GCD matrices satisfying the stronger property of being totally nonnegative (TN) or totally positive (TP). As we show, a GCD matrix is never TP when n ≥ 3, and is TN if and only if it is T N 2 , i.e., all its 2 × 2 minors are nonnegative. We next demonstrate that a GCD matrix is TN 2 if and only if the exponents of each prime divisor in the prime factorization of the x i s form a monotonic sequence. Reformulated in the language of kernels, our results characterize the subsets of integers over which the kernel K(x, y) = gcd(x, y) is totally nonnegative. The proofs of our characterizations depend on Gantmacher and Krein's notion of a Green's matrix. We conclude by showing that a GCD matrix is TN if and only if it is a Green's matrix. As a consequence, we obtain explicit formulas for all the minors and for the inverse of totally nonnegative GCD matrices.

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Section: Totally Nonnegative Kernelsmentioning

confidence: 99%

Total nonnegativity of GCD matrices and kernels

Guillot¹,

Wu²

2019

Linear Algebra and its Applications

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“…Our goal in this paper is to compute the simultaneous kernel for the entrywise powers of any given 3-PMP matrix; these comprise a much larger family of matrices than the cone P N (C) that was considered in [2]. As noted in the Introduction, the following Theorem is an important first step.…”

Section: The Hershkowitz-neumann-schneider Theorem and The Principal mentioning

confidence: 99%

“…Given a positive integer N and a subset I ⊂ C, let P N (I) denote the collection of N × N Hermitian positive semidefinite matrices with all entries in I. Motivated by the study of entrywise transformations of a matrix which preserve positivity, the authors computed in [2] the simultaneous kernel K(A) of Hadamard powers of a matrix A = (a ij ) ∈ P N (C): that is,…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, the simultaneous kernels of Hadamard powers are highly structured. Theorem 1.1 (see [2,Theorem 5.7]). Let A ∈ P N (C) \ {0} and let c 0 , .…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3, we identify precisely how principal minor positivity constrains the signature of a Hermitian matrix. In Section 4, we recall and extend to all Hermitian matrices the Schubert cell-type stratification of the cone of positive semidefinite matrices that was developed in [2]. Section 5 concludes the paper by classifying the simultaneous kernels of Hadamard powers of all 3-PMP matrices.…”

Section: Introductionmentioning

confidence: 99%

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Simultaneous kernels of matrix Hadamard powers

Belton

Guillot

Khare

et al. 2019

Linear Algebra and its Applications

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In previous work [Adv. Math. 298:325-368, 2016], the structure of the simultaneous kernels of Hadamard powers of any positive semidefinite matrix were described. Key ingredients in the proof included a novel stratification of the cone of positive semidefinite matrices and a well-known theorem of Hershkowitz, Neumann, and Schneider, which classifies the Hermitian positive semidefinite matrices whose entries are 0 or 1 in modulus. In this paper, we show that each of these results extends to a larger class of matrices which we term 3-PMP (principal minor positive).Date: March 26, 2018. 2010 Mathematics Subject Classification. 15B48 (primary); 15A21 (secondary).

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A Panorama of Positivity. I: Dimension Free

Belton¹,

Guillot²,

Khare³

et al. 2019

Trends in Mathematics

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This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included.

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Matrix positivity preservers in fixed dimension. I

Cited by 20 publications

References 51 publications

Total nonnegativity of GCD matrices and kernels

Total nonnegativity of GCD matrices and kernels

Simultaneous kernels of matrix Hadamard powers

A Panorama of Positivity. I: Dimension Free

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