Hadamard powers of some positive matrices

Jain, Tanvi

doi:10.1016/j.laa.2016.06.030

Cited by 17 publications

(19 citation statements)

References 15 publications

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“…In her 2017 paper [81], Jain provided a remarkable strengthening of the result mentioned at the end of the previous proof, which removes the dependence on t entirely. Theorem 6.2 (Jain [81]). Let…”

Section: Power Functionsmentioning

confidence: 74%

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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Section: Power Functionsmentioning

confidence: 74%

A Panorama of Positivity. I: Dimension Free

Belton¹,

Guillot²,

Khare³

et al. 2019

Trends in Mathematics

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“…. , x n are distinct positive real numbers and J n be the matrix of order n with each of its entries equals to 1, then X and J n are both ID, but their sum is not ID (see [12,Theorem 1.1]). The Cauchy matrix C = [c ij ] = 1 i+j , 1 ≤ i, j ≤ 3 is ID (see [1]), but its square C 2 is not ID because det(C 2 )…”

Section: By Theorem 8mentioning

confidence: 99%

“…If r > 0, then we denote the rth Hadamard power of a nonnegative matrix A = [a ij ] by A or (or (A) •r ), where A or = [a r ij ]. A lot of interest has been shown in studying the real entrywise powers preserving the positive semidefiniteness of various families of matrices, see [2,5,7,8,9,10,12,13]. A well-known result is that if A is a nonnegative PSD matrix of order n and r ≥ n − 2, then A •r is PSD.…”

mentioning

confidence: 99%

Positivity of Hadamard powers of a few band matrices

Panwar

Arikatla

2022

ELA

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Let $\mathbb{P}_G([0,\infty))$ and $\mathbb{P}_G^{'}([0,\infty))$ be the sets of positive semidefinite and positive definite matrices of order $n$, respectively, with nonnegative entries, where some positions of zero entries are restricted by a simple graph $G$ with $n$ vertices. It is proved that for a connected simple graph $G$ of order $n\geq 3$, the set of powers preserving positive semidefiniteness on $\mathbb{P}_G([0,\infty))$ is precisely the same as the set of powers preserving positive definiteness on $\mathbb{P}_G^{'}([0,\infty))$. In particular, this provides an explicit combinatorial description of the critical exponent for positive definiteness, for all chordal graphs. Using chain sequences, it is proved that the Hadamard powers preserving the positive (semi) definiteness of every tridiagonal matrix with nonnegative entries are precisely $r\geq 1$. The infinite divisibility of tridiagonal matrices is studied. The same results are proved for a special family of pentadiagonal matrices.

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“…The long history of this field starts with the Schur product theorem (see [5]), followed by the works of Schoenberg [9], Rudin [8] and others. In particular, the study of entrywise power functions x → x α has been of special interest to several mathematicians (see [1,2,3,4,6]). By the Schur product theorem, the mth Hadamard power A •m := [a m ij ] of any p.s.d.…”

Section: Introductionmentioning

confidence: 99%

“…They considered the matrix A ∈ P + n with (i, j)th entry 1 + ij and showed that if α is not an integer and 0 < α < n − 2, then A •α is not positive semi-definite for a sufficiently small positive number . Later, Jain [6] showed that this remains true if ij is replaced with x i x j for any distinct positive real numbers x 1 , . .…”

Section: Introductionmentioning

confidence: 99%

On Hadamard powers of positive semi-definite matrices

Baslingker¹,

Dan²

2022

Preprint

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Consider the set of scalars α for which the αth Hadamard power of any n × n positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the formwhat is the possible form of the set of such α for a fixed p.s.d. matrix with non-negative entries?". In all examples appearing in the literature, the set turns out to be union of a finite set and a semi-infinite interval. In this article, examples of matrices are given for which the set consists of a finite set and more than one disjoint interval of positive length. In fact, it is proved that for some matrices, the number of such disjoint intervals can be made arbitrarily large by taking n large. The case when the entries of the matrices are not necessarily non-negative is also considered.

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Hadamard powers of some positive matrices

Cited by 17 publications

References 15 publications

A Panorama of Positivity. I: Dimension Free

A Panorama of Positivity. I: Dimension Free

Positivity of Hadamard powers of a few band matrices

On Hadamard powers of positive semi-definite matrices

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