- Research
- Open Access
M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors
- Haitao Che^{1}Email author,
- Haibin Chen^{2} and
- Yiju Wang^{2}
https://doi.org/10.1186/s13660-019-1986-x
© The Author(s) 2019
- Received: 23 July 2018
- Accepted: 27 January 2019
- Published: 4 February 2019
Abstract
Inspired by symmetric Cauchy tensors, we define fourth-order partially symmetric Cauchy tensors with their generating vectors. In this article, we focus on the necessary and sufficient conditions for the M-positive semi-definiteness and M-positive definiteness of fourth-order Cauchy tensors. Moreover, the necessary and sufficient conditions of the strong ellipticity conditions for fourth-order Cauchy tensors are obtained. Furthermore, fourth-order Cauchy tensors are M-positive semi-definite if and only if the homogeneous polynomial for fourth-order Cauchy tensors is monotonically increasing. Several M-eigenvalue inclusion theorems and spectral properties of fourth-order Cauchy tensors are discussed. A power method is proposed to compute the smallest and the largest M-eigenvalues of fourth-order Cauchy tensors. The given numerical experiments show the effectiveness of the proposed method.
Keywords
- Cauchy tensor
- M-positive semi-definite
- M-positive definite
- Spectral property
- Homogeneous polynomial
- M-eigenvalue
- Power method
MSC
- 90C30
- 15A06
- 74B99
- 15A18
- 15A69
1 Introduction
Definition 1
Following the ideas of Cauchy matrix [2] and Cauchy tensor [6], we present the definition of fourth-order Cauchy tensors.
Definition 2
Recently, a lot of researchers have focused on structured tensors [6–28] such as M-tensors, Hankel tensors, Hilbert tensors, Cauchy tensors, completely positive tensors, B-tensors, and P-tensors. These papers not only gave some results on positive semi-definiteness property and spectral theory of structured tensors, but also revealed some important applications in data fitting and stochastic process [10, 29].
In this article, we focus on the M-positive semi-definiteness and M-positive definiteness conditions for fourth-order Cauchy tensors. Several spectral properties of M-positive semi-definite fourth-order Cauchy tensors are discussed. A power method is proposed to compute the smallest and the largest M-eigenvalues of fourth-order Cauchy tensors. In Sect. 2, the necessary and sufficient conditions for M-positive semi-definiteness and M-positive definiteness of fourth-order Cauchy tensors are obtained. Moreover, the necessary and sufficient conditions of the strong ellipticity condition for fourth-order Cauchy tensors are obtained. Furthermore, fourth-order Cauchy tensors are M-positive semi-definite if and only if the homogeneous polynomial of fourth-order Cauchy tensors is monotonically increasing in the nonnegative orthant of \(\mathbb{R}^{m}\times \mathbb{R}^{n}\), and the homogeneous polynomial is strictly monotone increasing when fourth-order Cauchy tensors are M-positive definite. In Sect. 3, several spectral inequalities are presented on the M-eigenvalue of fourth-order Cauchy tensors. We introduce a power method to compute the smallest and the largest M-eigenvalues of fourth-order Cauchy tensors, and numerical experiments show the effectiveness of the proposed method in Sect. 4.
At the end of the introduction, we make some notations that will be applied to the sequel. Denote vectors by lowercase boldface letters, i.e., x, y, … , and tensors are written to calligraphic capitals such as \(\mathcal{A}\), \(\mathcal{T}\), … . For \(\mathbf{x}=(x_{1},x_{2},\ldots ,x_{n})\), \({\mathbf{y}}=(y_{1},y_{2}, \ldots ,y_{n})\), \(\mathbf{x}\geq {\mathbf{y}}\) (\(\mathbf{x}\leq {\mathbf{y}}\)) means \(x_{i}\geq y_{i}\) (\(x_{i}\leq y _{i}\)) for all \(i\in [n]\).
2 M-positive semi-definiteness and M-positive definiteness of fourth-order Cauchy tensors
Now, we will show some necessary and sufficient conditions for fourth-order Cauchy tensors to be M-positive semi-definite.
Theorem 2.1
Let the vectors \(\mathbf{a}\in \mathbb{R}^{m}\), \(\mathbf{b}\in \mathbb{R}^{n} \) be generating vectors of the fourth-order Cauchy tensor \(\mathcal{C}\). Then the tensor \(\mathcal{C}\) is M-positive semi-definite if and only if \(a_{i}+b_{j}>0\) for all \(i\in [m]\), \(j\in [n]\).
Proof
From Theorem 2.1, we deduce the following corollary directly.
Corollary 2.1
Assume that the fourth-order Cauchy tensor \(\mathcal{C}\) and its generating vectors \(\mathbf{a}\in \mathbb{R}^{m}\), \(\mathbf{b}\in \mathbb{R}^{n}\) are defined in Theorem 2.1. Then the tensor \(\mathcal{C}\) is M-negative semi-definite if and only if \(a_{i}+b_{j}<0\) for all \(i\in [m]\), \(j\in [n]\).
Corollary 2.2
Assume that the fourth-order Cauchy tensor \(\mathcal{C}\) and its generating vectors \(\mathbf{a}\in \mathbb{R}^{m}\), \(\mathbf{b}\in \mathbb{R}^{n}\) are defined in Theorem 2.1. Then the tensor \(\mathcal{C}\) is not M-positive semi-definite if and only if there exist at least \(i\in [m]\), \(j\in [n]\), \(a_{i}+b_{j}<0\) holds.
Next, we will reveal some necessary and sufficient conditions for fourth-order Cauchy tensors to be M-positive definite.
Theorem 2.2
Assume that the vectors \(\mathbf{a}\in \mathbb{R}^{m}\), \(\mathbf{b} \in \mathbb{R}^{n} \) are generating vectors of the fourth-order Cauchy tensor \(\mathcal{C}\). For all \(i\in [m]\), \(j\in [n]\), if \(a_{i}+b_{j}>0\) and the elements of generating vectors a, b are mutually distinct, respectively, then the tensor \(\mathcal{C}\) is M-positive definite.
Proof
Moreover, the following conclusion shows that the conditions in Theorem 2.2 are necessary and sufficient conditions.
Theorem 2.3
Assume that the fourth-order Cauchy tensor \(\mathcal{C}\) and its generating vectors a, b are defined in Theorem 2.2. The tensor \(\mathcal{C}\) is M-positive definite if and only if \(a_{i}+b_{j}>0\) for all \(i\in [m]\), \(j\in [n]\), and the elements of generating vectors a, b are mutually distinct, respectively.
Proof
In what follows, we will give the definition of the monotonicity of a homogeneous polynomial with respect to fourth-order Cauchy tensors.
For any \(\mathbf{x},\overline{\mathbf{x}} \in \mathbb{R}^{m}\) and \(\mathbf{y},\overline{\mathbf{y}} \in \mathbb{R}^{n}\), if \(f( \mathbf{x},\mathbf{y})\geq f(\overline{\mathbf{x}},\overline{ \mathbf{y}})\) when \(\mathbf{x}\geq \overline{\mathbf{x}}\) and \(\mathbf{y}\geq \overline{\mathbf{y}}\), (\(\mathbf{x}\leq \overline{ \mathbf{x}}\) and \(\mathbf{y}\leq \overline{\mathbf{y}}\)), then \(f(\mathbf{x},\mathbf{y})\) is called monotonically increasing (monotonically decreasing respectively). If \(f(\mathbf{x},\mathbf{y})> f(\overline{\mathbf{x}},\overline{\mathbf{y}})\) when \(\mathbf{x} \geq \overline{\mathbf{x}}\), \(\mathbf{x}\neq \overline{\mathbf{x}}\) and \(\mathbf{y}\geq \overline{\mathbf{y}}\), \(\mathbf{y}\neq \overline{ \mathbf{y}}\) (\(\mathbf{x}\leq \overline{\mathbf{x}}\), \(\mathbf{x}\neq \overline{ \mathbf{x}}\) and \(\mathbf{y}\leq \overline{\mathbf{y}}\), \(\mathbf{y} \neq \overline{\mathbf{y}}\)), then \(f(\mathbf{x},\mathbf{y})\) is called strictly monotone increasing (strictly monotone decreasing respectively).
The following conclusions reveal the relationships between M-positive semi-definiteness of fourth-order Cauchy tensor and the monotonicity of a homogeneous polynomial with respect to the proposed Cauchy tensor.
Theorem 2.4
Let \(\mathcal{C}\) be a fourth-order Cauchy tensor with generating vectors \(\mathbf{a}\in \mathbb{R}^{m}\) and \(\mathbf{b}\in \mathbb{R} ^{n}\). Then the tensor \(\mathcal{C}\) is M-positive semi-definite if and only if the homogeneous polynomial \(f(\mathbf{x},\mathbf{y})\) is monotonically increasing in \(\mathbb{R}^{m}_{+}\times \mathbb{R}^{n} _{+}\).
Proof
Theorem 2.5
Let \(\mathcal{C}\) be a fourth-order Cauchy tensor with generating vectors \(\mathbf{a}\in \mathbb{R}^{m}\) and \(\mathbf{b}\in \mathbb{R} ^{n}\). If the tensor \(\mathcal{C}\) is M-positive definite, then the homogeneous polynomial \(f(\mathbf{x},\mathbf{y})\) is strictly monotone increasing in \(\mathbb{R}^{m}_{+}\times \mathbb{R}^{n}_{+}\).
Proof
Now, we are in a position to propose an example to reveal that the strictly monotone increasing property for the polynomial \(f( \mathbf{x},\mathbf{y})\) is only a necessary condition for the M-positive definiteness property of the tensor \(\mathcal{C}\) but not a sufficient condition.
Example 2.1
Applying the above definitions, we have the following technical conclusion.
Theorem 2.6
- (i)
the tensor \(\mathcal{C}\) is copositive;
- (ii)
for every \(\mathbf{q}>\mathbf{0}\), \(\operatorname{TCP}(\mathbf{q},\mathcal{C})\) has a unique solution;
- (iii)for every index set \(N \subset [n]\), the systemhas no solution, where \(\mathbf{x}^{N} \in \mathbb{R}^{|N|} \);$$ \mathcal{C}^{\mid N\mid }\bigl(\mathbf{x}^{N}\bigr)^{3}< \mathbf{0}, \quad {\mathbf{x}} ^{N}\geq \mathbf{0} $$
- (iv)
for all \(i\in [n]\), \(a_{i}+b_{i}>0\).
Proof
(ii) ⇒ (iii). Following the proof of Theorem 3.1 [34], we have the desired result.
(iv) ⇒ (i). By Theorem 2.1, we obtain that the tensor \(\mathcal{C}\) is M-positive semi-definite, which means that the tensor \(\mathcal{C}\) is copositive. □
3 Spectral properties for fourth-order Cauchy tensors
In this section, we discuss M-eigenvalue inclusion theorems and spectral properties of fourth-order Cauchy tensors. M-eigenvalue problem has a close relationship with the strong ellipticity condition, which plays an important role in nonlinear elasticity and in materials, since it can ensure an elastic material to satisfy some mechanical properties. Thus, to identify whether the strong ellipticity condition of a given material holds or not becomes an important problem in mechanics [29, 35–38].
Theorem 3.1
([39])
The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive.
From Theorems 2.3 and 3.1, we can obtain the necessary and sufficient conditions of the strong ellipticity condition for fourth-order Cauchy tensors.
Theorem 3.2
Let the vectors \(\mathbf{a}\in \mathbb{R}^{n}\), \(\mathbf{b}\in \mathbb{R}^{n} \) be generating vectors of the fourth-order Cauchy tensor \(\mathcal{C}\). The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the tensor \(\mathcal{C}\) is positive.
Theorem 3.3
Let the vectors \(\mathbf{a}\in \mathbb{R}^{n}\), \(\mathbf{b}\in \mathbb{R}^{n} \) be generating vectors of the fourth-order Cauchy tensor \(\mathcal{C}\). The strong ellipticity condition holds if and only if \(a_{i}+b_{j}>0\) for all \(i, j\in [n]\), and the elements of generating vectors a, b are mutually distinct, respectively.
Now, inspired by the idea of H-eigenvalue inclusion theorem [40], we establish the following M-eigenvalue inclusion theorems for fourth-order Cauchy tensors.
Theorem 3.4
Proof
Theorem 3.5
Proof
Next, we will reveal several spectral properties for fourth-order Cauchy tensors.
Theorem 3.6
Suppose that the tensor \(\mathcal{C}\) is a fourth-order Cauchy tensor with generating vectors \(\mathbf{a}\in \mathbb{R}^{m}\), \(\mathbf{b} \in \mathbb{R}^{n} \), and for all \(i\in [m]\), \(j\in [n]\) such that \(a_{i}+b_{j}>0\). Then the tensor \(\mathcal{C}\) is M-positive definite if and only if its M-eigenvalues are positive.
Proof
Theorem 3.7
Suppose that the fourth-order Cauchy tensor \(\mathcal{C}\) and its generating vectors a, b are defined as in Theorem 3.6 with the elements of generating vectors a, b mutually distinct, respectively. If \(\lambda \in \sigma ( \mathcal{C})\) is an M-eigenvalue of the tensor \(\mathcal{C}\) with non-negative left M-eigenvector x or non-negative right M-eigenvector y, then \(\lambda \neq 0\).
Proof
4 Power method of fourth-order Cauchy tensors
In this section, a power method is proposed to compute the smallest and the largest M-eigenvalues of fourth-order Cauchy tensors. It is well known that the power method is an efficient method to solve the largest eigenvalue of a matrix [41]. The method has successfully extended to compute the largest Z-eigenvalue in magnitude of higher-order tensors [40] and the largest M-eigenvalue of a fourth-order partially symmetric tensor [42]. Motivated by these, we first propose a power method to compute the smallest M-eigenvalue of fourth-order Cauchy tensors.
Now, we are in a position to propose a power method to compute the smallest M-eigenvalue of a fourth-order Cauchy tensor \(\mathcal{C }\).
Algorithm 4.1
Initialization step: Choose initial points \(\mathbf{x}_{0} \in \mathbb{R}^{m}\), \(\mathbf{y}_{0} \in \mathbb{R}^{n}\), and let \(k=0\).
Final step: Output the smallest M-eigenvalue \(\alpha -\overline{f}( \mathbf{x}^{*}, \mathbf{y}^{*})\) of the tensor \(\mathcal{{\mathcal{C}}}\) and the associated M-eigenvectors \(\mathbf{x}^{*}\), \(\mathbf{y}^{*}\).
The following numerical experiments show the effectiveness of the proposed method. The whole program was written in Matlab 7.0. All the numerical results were carried out on a personal Lenovo Thinkpad computer with Intel(R) Core(TM) i7-6500U CPU 2.50 GHz and RAM 8.00 GB. In the implementation, we choose \(\|x_{k+1}-x_{k}\|+\|y_{k+1}-y_{k}\| \leq 10^{-10}\) as the stopping criterion, and take the parameter \(\varepsilon =0.0001\).
Example 4.1
Example 4.2
Example 4.3
Consider a fourth-order Cauchy tensor \(\mathcal{C}\) with generating vectors \(\mathbf{a}=\operatorname{rand}(20,1)+10\) and \(\mathbf{b}=30*\operatorname{rand}(30,1) \). The variation of the objective function value corresponding to the tensor \(\mathcal{C}\) during the iteration process can be shown in Fig. 4. For the tensor \(\mathcal{C}\), its smallest M-eigenvalue is 0.0594 and the largest M-eigenvalue is 12.4184.
Example 4.4
Consider a fourth-order Cauchy tensor \(\mathcal{C}\) with generating vectors \(\mathbf{a}=5*\operatorname{rand}(30,1)+10\) and \(\mathbf{b}=\operatorname{rand}(40,1)+8 \). The variation of the objective function value corresponding to the tensor \(\mathcal{C}\) during the iteration process can be shown in Fig. 4. For the tensor \(\mathcal{C}\), its smallest M-eigenvalue is 0.0229 and the largest M-eigenvalue is 28.9232.
5 Final remarks
In this article, the necessary and sufficient conditions for the M-positive semi-definiteness and M-positive definiteness of fourth-order Cauchy tensors are discussed. Moreover, the necessary and sufficient conditions of the strong ellipticity condition for fourth-order Cauchy tensors are obtained. Furthermore, we reveal that fourth-order Cauchy tensors are M-positive semi-definite if and only if there is a monotone increasing homogeneous polynomial defined in the nonnegative orthant of \(\mathbb{R}^{m}\times \mathbb{R}^{n}\). Several M-eigenvalue inclusion theorems and spectral properties of fourth-order Cauchy tensors are discussed. A power method is proposed to compute the smallest and the largest M-eigenvalues of fourth-order Cauchy tensors. The given numerical experiments show the effectiveness of the proposed method.
However, there are still some questions that we are not sure about now. Can we have the type of Cauchy–Toeplitz tensors with the partially symmetric property? If so, how about their spectral properties? What are the necessary and sufficient conditions for their M-positive semi-definiteness?
Declarations
Acknowledgements
Haibin Chen’s research was done during his postdoctoral period in Qufu Normal University. We are thankful to Dr. Min Sun for his valuable comments to the numerical experiments.
Funding
This work was supported by the Natural Science Foundation of China (11401438, 11671228, 11601261, 11571120), Shandong Provincial Natural Science Foundation (ZR2016AQ12), and Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J14LI52).
Authors’ contributions
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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