Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors

Chen, Haibin; Qi, Liqun

doi:10.3934/jimo.2015.11.1263

Cited by 54 publications

(40 citation statements)

References 26 publications

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“…Symmetric Cauchy tensors was first studied in [2]. Some checkable sufficient and necessary conditions for an even order symmetric Cauchy tensor to be positive semi-definite or positive definite were provided in [2], which extends the matrix cases established in [10]. Let c = (c 1 , c 2 , · · · , c n ) T ∈ R n with c i1 + c i2 + · · · + c im = 0 for all i j ∈ {1, .…”

Section: Characterizing Sos Decomposition For Even Order Cauchy Tensorsmentioning

confidence: 99%

SOS tensor decomposition: Theory and applications

Chen

2016

Communications in Mathematical Sciences

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In this paper, we examine structured tensors which have sum-of-squares (SOS) tensor decomposition, and study the SOS-rank of SOS tensor decomposition. We first show that several classes of even order symmetric structured tensors available in the literature have SOS tensor decomposition. These include positive Cauchy tensors, weakly diagonally dominated tensors, B0-tensors, double B-tensors, quasi-double B0-tensors, M B0-tensors, H-tensors, absolute tensors of positive semi-definite Z-tensors and extended Z-tensors. We also examine the SOS-rank of SOS tensor decomposition and the SOS-width for SOS tensor cones. The SOS-rank provides the minimal number of squares in the SOS tensor decomposition, and, for a given SOS tensor cone, its SOS-width is the maximum possible SOS-rank for all the tensors in this cone. We first deduce an upper bound for general tensors that have SOS decomposition and the SOS-width for general SOS tensor cone using the known results in the literature of polynomial theory. Then, we provide an explicit sharper estimate for the SOS-rank of SOS tensor decomposition with bounded exponent and identify the SOS-width for the tensor cone consisting of all tensors with bounded exponent that have SOS decompositions. Finally, as applications, we show how the SOS tensor decomposition can be used to compute the minimum H-eigenvalue of an even order symmetric extended Z-tensor and test the positive definiteness of an associated multivariate form. Numerical experiments are also provided to show the efficiency of the proposed numerical methods ranging from small size to large size numerical examples.

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Section: Characterizing Sos Decomposition For Even Order Cauchy Tensorsmentioning

confidence: 99%

SOS tensor decomposition: Theory and applications

Chen

2016

Communications in Mathematical Sciences

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“…, 1/(N + 1)). Recently, some progress has been made on the study of other kinds of structured tensors such as Cauchy tensor, P-tensor and B-tensor, and Centro-Symmetric tensor [12,2,17,18,3].…”

Section: Introductionmentioning

confidence: 99%

Hankel tensors, Vandermonde tensors and their positivities

2016

Linear Algebra and its Applications

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“…Clearly, an m-order n-dimensional Hilbert tensor is a Hankel tensor with v = (1, 1 2 , 1 3 , · · · , 1 nm ), introduced by Qi [19]. Also see Chen and Qi [3], Xu [26] for more details of Hankel tensors. Hilbert tensor (hypermatrix) is a natural extension of Hilbert matrix, which was introduced by Hilbert [7].…”

Section: Introductionmentioning

confidence: 99%

Infinite and finite dimensional generalized Hilbert tensors

Mei

Song

2017

Linear Algebra and its Applications

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In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a},\ a\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=1,2,\cdots,n, $$ and show that its $H$-spectral radius and its $Z$-spectral radius are smaller than or equal to $M(a)n^{m-1}$ and $M(a)n^{\frac{m}{2}}$, respectively, here $M(a)$ is a constant only dependent on $a$. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for $a\geq1$. For an $m$-order infinite dimensional generalized Hilbert tensor $\mathcal{H}_{\infty}$ with $a>0$, we prove that $\mathcal{H}_{\infty}$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^{1}$ into $l^{p}\ (1

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Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors

Cited by 54 publications

References 26 publications

SOS tensor decomposition: Theory and applications

SOS tensor decomposition: Theory and applications

Hankel tensors, Vandermonde tensors and their positivities

Infinite and finite dimensional generalized Hilbert tensors

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