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Some inequalities for the Fan product of M-tensors
Journal of Inequalities and Applications volume 2018, Article number: 257 (2018)
Abstract
In this paper, we investigate some inequalities for the Fan product of M-tensors. We propose exact characterizations of M-tensors and establish some inequalities on the minimum eigenvalue for the Fan product of two M-tensors. Furthermore, the inclusion relations among them are discussed. Numerical examples show the validity of the conclusions.
1 Introduction
Let \(\mathbb{C} (\mathbb{R})\) be the set of all complex (real) numbers, \(\mathbb{R}_{+} (\mathbb{R}_{++})\) be the set of all nonnegative (positive) numbers, \(\mathbb{C}^{n} (\mathbb{R}^{n})\) be the set of all dimension n complex (real) vectors, and \(\mathbb{R}_{+}^{n} (\mathbb{R}_{++}^{n})\) be the set of all dimension n nonnegative (positive) vectors. An mth order n-dimensional tensor \(\mathcal{A}=(a_{i_{1}i_{2}\dots i_{m}})\) is a higher-order generalization of matrices, which consists of \(n^{m}\) entries:
\(\mathcal{A}\) is called nonnegative (positive) if \(a_{{i_{1}}{i_{2}}\ldots {i_{m}}}\in\mathbb{R}_{+}\) (\(a_{{i_{1}}{i_{2}}\ldots{i_{m}}}\in\mathbb{R}_{++}\)).
Tensors have many similarities with matrices and many related results of matrices such as determinant, eigenvalue, and algorithm theory can be extended to higher order tensors [1–3]. Furthermore, structured matrices such as nonnegative matrices, H-matrices and M-matrices can also be extended to higher order tensors and these are becoming the focus of recent tensor research [4–26]. In particular, M-tensors play important roles in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control [27–29] and spectral hypergraph theory [3, 30, 31].
On the other hand, Fan product of M-matrices and Hadamard product of nonnegative matrices are significant for practical problems, such as the weak minimum principle in partial differential equations, products of integral equation kernels, characteristic functions in probability theory, the study of association schemes in combinatorial theory, and so on (see [32]). Some inequalities on the spectral radius for the Hadamard product of two nonnegative matrices and some inequalities on the minimum eigenvalue for the Fan product of two M-matrices can be found in [33–37]. Recently, Sun et al. [14] investigated some inequalities for the Hadamard product of tensors and obtained some bounds on the spectral radius, and used them to estimate the spectral radius of a directly weighted hypergraph. It is well known that an M-tensor is defined based on a Z-tensor and its algebra properties can be explored using the spectral theory of nonnegative tensors [23]. Motivated by these observations, we expect to establish sharp lower bounds on the minimal eigenvalue for the Fan product of two M-tensors and discuss some inclusion relations among them.
The remaining of this paper is organized as follows. In Sect. 2, we introduce important notation and recall some preliminary results on tensor analysis. In Sect. 3, based on exact characterizations of M-tensors, we give a lower bound on the minimum eigenvalue for the Fan product of two M-tensors. An improved result is established for irreducible nonnegative tensors by the ratio of the smallest and largest values of a Perron vector. Finally, making use of the information of the absolute maximum in the off-diagonal elements, we obtain a new lower bound on the minimum eigenvalue for the Fan product. With numerical examples, we exhibit the efficiency of the results given in Theorems 1–3.
2 Notation and preliminaries
We start this section with some fundamental notions and properties developed in tensor analysis [1, 3], which are needed in the subsequent analysis.
Definition 1
Let \(\mathcal{A}\) be an m-order n-dimensional tensor. Assume that \(\mathcal{A}x^{m-1}\) is not identical to 0. We say that \((\lambda,x)\in\mathbb{C}\times(\mathbb{C}^{n}\setminus \{0\})\) is an eigenvalue–eigenvector of \(\mathcal{A}\) if
where \((\mathcal{A} x^{m-1})_{i}=\sum^{n}_{i_{2},\ldots, i_{m}=1}a_{{i}{i_{2}}\dots{i_{m}}}x_{i_{2}}\cdots x_{i_{m}}\), \(x^{[m-1]}=[x^{m-1}_{1},x^{m-1}_{2},\ldots,x^{m-1}_{n}]^{T}\), and \((\lambda,x)\) is called an H-eigenpair if they are both real.
Definition 2
Let \(\mathcal{A}\) and \(\mathcal{I}\) be m-order n-dimensional tensors.
-
(i)
We call \(\sigma(\mathcal{A})\) as the set of all eigenvalues of \(\mathcal{A}\). Assume \(\sigma(\mathcal{A})\neq\emptyset\). Then the spectral radius of \(\mathcal{A}\) is denoted by
$$\rho(\mathcal{A})=\max\bigl\{ |\lambda|: \lambda\in\sigma(\mathcal{A})\bigr\} . $$Meanwhile, we use \(\tau(\mathcal{A})\) to denote the minimal value of the real part of eigenvalues of \(\mathcal{A}\).
-
(ii)
We call a tensor \(\mathcal{A}\) reducible if there exists a nonempty proper index subset \(I\subset\{1,2,\dots,n\}\) such that
$$a_{{i_{1}}{i_{2}}\dots{i_{m}}}=0, \quad \forall i_{1}\in I, i_{2}, \dots,i_{m}\notin I. $$If \(\mathcal{A}\) is not reducible, then we call \(\mathcal{A}\) irreducible.
-
(iii)
We call a nonnegative matrix \(GM(\mathcal{A})\) the representation associated to a nonnegative tensor \(\mathcal{A}\), if the \((i, j)\)th entry of \(GM(\mathcal{A})\) is defined to be the sum of \(a_{ii_{2} i_{3}\ldots i_{m}}\) with indices \(j\in\{i_{2}, i_{3}, \ldots, i_{m}\} \). We call a tensor \(\mathcal{A}\) weakly reducible, if its representation \(GM(\mathcal{A})\) is reducible. It is weakly irreducible if it is not weakly reducible.
-
(iv)
We call \(\mathcal{I}\) is a unit tensor whose entries are
$$\delta_{{i_{1}}{i_{2}}\ldots{i_{m}}}= \textstyle\begin{cases} 1, & \text{if } i_{1}=i_{2}= \cdots=i_{m}, \\ 0, & \text{otherwise}. \end{cases} $$
It is noted that the spectral radius \(\rho(\mathcal{A})\) is the largest H-eigenvalue for the nonnegative tensor [4] and \(\tau (\mathcal{A})\) is smallest H-eigenvalue for the M-tensor [23].
The Perron–Frobenius theorem for nonnegative weakly irreducible tensors has been established in [9, 11, 22].
Lemma 1
Let \(\mathcal{A}\) be a weakly irreducible nonnegative tensor of order m and dimension n. Then the following results hold:
-
(i)
\(\mathcal{A}\) has a positive eigenpair \((\lambda,x)\) and x is unique up to a multiplicative constant.
-
(ii)
$$\min_{x\in\mathbb{R}^{n}_{++}}\max_{1\leq i\leq n}\frac{(\mathcal {A}x^{m-1})_{i}}{{{x_{i}}}^{[m-1]}}=\rho( \mathcal{A})=\max_{x\in\mathbb {R}^{n}_{+}\setminus \{0\}}\min_{x_{i}\neq0, 1\leq i\leq n} \frac{(\mathcal {A}x^{m-1})_{i}}{{{x_{i}}}^{[m-1]}}. $$
The following specially structured tensors are extended from matrices [8, 23].
Definition 3
Let \(\mathcal{A}\) and \(\mathcal{U}\) be m-order n-dimensional tensors.
-
(i)
We call \(\mathcal{A}\) is a Z-tensor if all its off-diagonal entries are nonpositive.
-
(ii)
We call \(\mathcal{A}\) is an M-tensor if there exist a nonnegative tensor \(\mathcal{U}\) and a positive real number \(\eta\geq \rho(\mathcal{U})\) such that
$$\mathcal{A}=\eta\mathcal{I}-\mathcal{U}. $$If \(\eta> \rho(\mathcal{U})\), then \(\mathcal{A}\) is called a strong M-tensor.
-
(iii)
We call \(\mathcal{A}\) is a weakly irreducible M-tensor if \(\mathcal{U}\) is weakly irreducible.
-
(iv)
Assume \(\mathcal{A}\) and \(\mathcal{B}\) are M-tensors. The Fan product of \(\mathcal{A}\) and \(\mathcal{B}\) is denoted by \(\mathcal {A}\star\mathcal{B}=\mathcal{D}=(d_{i_{1} i_{2}\ldots i_{m}})\) and defined by
$${d_{{i_{1}}{i_{2}} \ldots{i_{m}}}} = { \textstyle\begin{cases} {{a_{i \ldots i}}{b_{i \ldots i}}}, & {{i_{1}} = {i_{2}} = \cdots={i_{m}} = i}, \\ { - |{a_{{i_{1}}{i_{2}} \ldots{i_{m}}}}{b_{{i_{1}}{i_{2}} \ldots{i_{m}}}}|}, & \text{otherwise}. \end{cases}\displaystyle } $$
It is easy to see that all the diagonal entries of an M-tensor are nonnegative [23], and the (strong) M-tensor is closely linked with the diagonal dominance defined below.
Definition 4
An m-order n-dimensional tensor \(\mathcal{A}\) is called diagonally dominant if
\(\mathcal{A}\) is called strictly diagonally dominant if the strict inequalities hold for all \(i\in N\).
Define a positive diagonal matrix \(D =\operatorname{diag}(d_{1},\ldots, d_{n})\) and set
We obtain the following necessary and sufficient condition for identifying M-tensors.
Lemma 2
([23])
Suppose \(\mathcal{A}\) is a weakly irreducible Z-tensor and its all diagonal elements are nonnegative. Then \(\mathcal{A}\) is an (strong) M-tensor if and only if there exists a positive diagonal matrix D such that \(\mathcal{B}\) defined in (1) is (strictly) diagonally dominant.
3 Some inequalities on the minimum eigenvalue for the Fan product
In this section, we shall give lower bounds on the minimum eigenvalue for the Fan product. Firstly, we establish characterizations of M-tensors.
Lemma 3
Let \(\mathcal{Q}\) be a weakly irreducible M-tensor of order m and dimension n. If \(\mathcal{Q}z^{m-1}\geq kz^{[m-1]}\) for a vector \(z\in R^{n}_{++}\) and a real number k, then \(k\leq\tau(\mathcal{Q})\).
Proof
Since \(\mathcal{Q}\) is an M-tensor, there exists a nonnegative tensor \(\mathcal{U}\) such that
where λ is a nonnegative real number and \(\lambda\geq\rho (\mathcal{U})\). It is easy to see that \(\tau(\mathcal{Q})=\lambda-\rho (\mathcal{U})\). Furthermore, \(\rho(\mathcal{U})=\lambda-\tau(\mathcal{Q})\). Taking into account that \(\mathcal{Q}\) is weakly irreducible, we deduce that \(\mathcal{U}\) is weakly irreducible. From the assumption and (2), we have
that is,
It follows from Lemma 1 that
So, \(\tau(\mathcal{Q})\geq k\). □
Lemma 4
Let \(\mathcal{P}\), \(\mathcal{Q}\) be two M-tensors of order m and dimension n. Then \(\mathcal{P}\star\mathcal{Q}\) is an M-tensor. Furthermore, if \(\mathcal{P}\), \(\mathcal{Q}\) are strong M-tensors, then \(\mathcal{P}\star\mathcal{Q}\) is a strong M-tensor.
Proof
By the definition of \(\mathcal{P}\star\mathcal{Q}\), it holds that
Since \(\mathcal{P}\), \(\mathcal{Q}\) are M-tensors, by Lemma 1, there exist positive diagonal matrices C, D such that
with
Specifically,
Taking into account that \(\mathcal{A}\) and \(\mathcal{B}\) are diagonally dominant, we conclude that
Furthermore, it holds that
Hence, it follows from (3) that there exists a positive diagonal matrix \(U=\operatorname{diag} (c_{1} d_{1},c_{2} d_{2}, \ldots,c_{n} d_{n})\) such that
It follows from Lemma 2 that \(\mathcal{P}\star\mathcal{Q}\) is an M-tensor. By a similar argument as for the first conclusion, we can obtain the second conclusion. □
Suppose that \(\mathcal{P}=(p_{i_{1} i_{2} \ldots i_{m}})\) is a strong M-tensor of order m and dimension n. Set \(\mathcal{N}=\mathcal {D}-\mathcal{P}\), where \(\mathcal{D}\) denotes the diagonal tensor of the same order, dimension and diagonal entries as \(\mathcal{P}\). Note that \(p_{ii\ldots i}>0\) for \(i\in N\) when \(\mathcal{P}\) is a strong M-tensor. Define \(J_{\mathcal{P}}=\mathcal{D}^{-1}\mathcal{N}\). Obviously, \(J_{\mathcal{P}}\) is nonnegative. The following result characterizes \(J_{\mathcal{P}}\) in terms of the spectral radius.
Lemma 5
Suppose that \(\mathcal{P}=(p_{i_{1} i_{2} \ldots i_{m}})\) is a strong M-tensor of order m and dimension n. Then
Furthermore, if \(\mathcal{P}\) is weakly irreducible, then
Proof
Let \(\mathcal{P}=(p_{i_{1} i_{2} \ldots i_{m}})\) be a strong M-tensor. Then there exists a positive vector \(u= (u_{i})\) such that
that is,
Since the tensor \(J_{\mathcal{P}}\) is nonnegative, by Lemma 1 and (4), we have
Furthermore, \(J_{\mathcal{P}}\) is weakly irreducible when \(\mathcal{P}\) is weakly irreducible. From Lemma 1 and (4), it holds that
□
The following example shows that the bound of Lemma 5 is tight.
Example 1
Let \(\mathcal{P}=(p_{ijk})\) be a tensor of order 3 and dimension 3 with elements defined as follows:
By computations, we get \(\tau(\mathcal{P})=1\) and
Based on the characterizations of M-tensors, we can immediately obtain these bounds from the following result.
Theorem 1
If \(\mathcal{P}\) and \(\mathcal{Q}\) are two strong M-tensors of order m and dimension n, then
Proof
Let us distinguish two cases.
Case 1. \(\mathcal{P}\) and \(\mathcal{Q}\) are both weakly irreducible. It follows from Lemma 4 that \(\mathcal{P}\star\mathcal{Q}\) is a strong M-tensor. Since \(J_{\mathcal{P}}\) and \(J_{\mathcal{Q}}\) are weakly irreducible nonnegative tensors, from Lemma 1, there exist two positive vectors u, v such that
equivalently,
Let \(z=(z_{i})\), where \(z_{i}= u_{i} v_{i} \in\mathbb{R}_{++}\) for \(i\in N\). Setting \(\mathcal{U}=\mathcal{P} \star\mathcal{Q}\), for \(i \in N\), we obtain
It follows from Lemma 3 and (9) that
Case 2. Either \(\mathcal{P}\) or \(\mathcal{Q}\) is weakly reducible. Let \(\mathcal{S}\) be a tensor of order m and dimension n with
Then both \(\mathcal{P}- \epsilon\mathcal{S}\) and \(\mathcal{Q}- \epsilon\mathcal{S}\) are weakly irreducible tensors for any \(\epsilon>0\). Now, we claim that \(\mathcal{P}- \epsilon\mathcal{S}\) and \(\mathcal {Q}- \epsilon\mathcal{S}\) are both strong M-tensors when \(\epsilon >0\) is sufficiently small. Since \(\mathcal{P}\) and \(\mathcal{Q}\) are strong M-tensors, there exist positive diagonal matrices C, D such that
with
In particular,
By Lemma 2, one has
Set
and
Then for any \(0<\epsilon<\epsilon_{0}\), it holds that \(\mathcal {P}-\epsilon\mathcal{S}\) and \(\mathcal{Q}-\epsilon\mathcal{S}\) are strong M-tensors. Substituting \(\mathcal{P}-\epsilon\mathcal{S}\) and \(\mathcal {Q}-\epsilon\mathcal{S}\) for \(\mathcal{P}\) and \(\mathcal{Q}\) and letting \(\epsilon\to0\), we obtain the desired results by the continuity of \(\tau(\mathcal{P}-\epsilon\mathcal{S})\) and \(\tau (\mathcal{Q}-\epsilon\mathcal{S})\). □
Next, we give a lemma about the ratio of the smallest and largest values of a Perron vector for an irreducible nonnegative tensor.
Lemma 6
(Lemma 3.2 of [35])
Let \(\mathcal{B}\) be a nonnegative irreducible tensor of order \(m\geq3\) and dimension n with a Perron vector y. Then we have
where \(\kappa(\mathcal{B})=\max_{2\leq k,k'\leq m} \min_{\substack{1\leq i_{1},i_{1'} \leq n\\1\leq i_{k}=i_{k'} \leq n}}\frac{\sum^{n}_{{\underbrace{i_{2},\ldots,i_{m}}_{\textit{except }i_{k}}}}b_{i_{1} i_{2}\ldots i_{m}}}{ \sum^{n}_{{\underbrace{i_{2'},\ldots,i_{m'}}_{\textit{except } i_{k'}}}}b_{i_{1'} i_{2'}\ldots i_{m'}}}\).
Based on the above lemma, we propose the following theorem, which provides a sharp bound under the condition of irreducibility.
Theorem 2
Suppose that \(\mathcal{P}\) and \(\mathcal{Q}\) are two irreducible strong M-tensors of order m and dimension n, and \(\rho(J_{\mathcal{P}})\) and \(\rho (J_{\mathcal{Q}})\) are their spectral radii with eigenvalue vectors u and v, respectively. Then,
where \(\alpha=\kappa(J_{P})^{\frac{m-1}{2}}\leq [\frac{u_{\min}}{u_{\max }}]^{(m-1)}\), \(\beta=\kappa(J_{Q})^{\frac{m-1}{2}}\leq[\frac{v_{\min }}{v_{\max}}]^{(m-1)}\), \(r^{j}_{i}(J_{\mathcal{P}})=\sum_{\substack{ \delta_{{i}{i_{2}}\ldots{i_{m}}}=0\\ \delta_{{j}{i_{2}}\ldots{i_{m}}}=0}} \frac{|p_{i{i_{2}}\ldots{i_{m}}}|}{p_{i\ldots i}}\) and \(r^{j}_{i}(J_{\mathcal{Q}})=\sum_{\substack{ \delta_{{i}{i_{2}}\ldots{i_{m}}}=0\\ \delta_{{j}{i_{2}}\ldots{i_{m}}}=0}}\frac{|q_{i{i_{2}}\ldots{i_{m}}}|}{q_{i\ldots i}}\).
Proof
It follows from Lemma 4 that \(\mathcal{P}\star \mathcal{Q}\) is a strong M-tensor. Since \(\mathcal{P}\) and \(\mathcal{Q}\) are strongly irreducible M-tensors, \(J_{\mathcal{P}}\) and \(J_{\mathcal{Q}}\) are irreducible nonnegative tensors. By the assumption that \(\rho(J_{\mathcal{P}})\) and \(\rho(J_{\mathcal{Q}})\) are the spectral radii with eigenvalue vectors u and v, we deduce that u and v are positive vectors such that
equivalently,
Let \(z=(z_{i})\), where \(z_{i}= u_{i} v_{i}\in\mathbb{R}_{++}\) for \(i\in N\). Setting \(\mathcal{U} = \mathcal{P} \star\mathcal{Q}\), for \(i \in N\), by (10) and (11), we have
From (10) and Lemma 6, we deduce
Similarly,
Combining (12) with (13) and (14), we have
It follows from (15) and Lemma 3 that
□
Remark 1
The bound in Theorem 2 is sharper than the result of Theorem 1, since \(\frac{\alpha\beta|p_{i j\ldots j}|}{ p_{i\ldots i}}r^{j}_{i}(J_{\mathcal {Q}})+\frac{\alpha\beta|q_{i j\ldots j}|}{ q_{i\ldots i}} r^{j}_{i}(J_{\mathcal{P}})\geq0\).
The following example exhibits the efficiency of Theorems 1 and 2.
Example 2
Let \(\mathcal{P}=(p_{ijk})\), \(\mathcal{Q}=(q_{ijk})\) be two tensors of order 3 and dimension 3 with elements defined as follows:
where
It is clear that \(\min_{1\leq i\leq n}(p_{i\ldots i}q_{i\ldots i})=9\). By computations, we get
From Theorem 1, we have
According to Theorem 2, we obtain
By making use of the information of the absolute maximum in the off-diagonal elements, we are at the position to establish the following theorem.
Theorem 3
Suppose that \(\mathcal{P}\) and \(\mathcal{Q}\) are two strong M-tensors of order m and dimension n and assume that \(\rho(J_{\mathcal{P}})\) and \(\rho(J_{\mathcal{Q}})\) are the corresponding spectral radii. Then
where \(\alpha_{i}=\max_{\delta_{{i}{i_{2}}\ldots{i_{m}}}=0}|p_{i i_{2}\ldots i_{m}}|\) and \(\beta_{i}=\max_{\delta_{{i}{i_{2}}\ldots{i_{m}}}=0}|q_{i i_{2}\ldots i_{m}}|\).
Proof
The proof is broken into two cases.
Case 1. \(\mathcal{P}\) and \(\mathcal{Q}\) are both weakly irreducible. It follows from Lemma 4 that \(\mathcal{P}\star\mathcal{Q}\) is a strong M-tensor. Since \(J_{\mathcal{P}}\) and \(J_{\mathcal{Q}}\) are weakly irreducible nonnegative tensors, by Lemma 1, there exist two positive eigenvectors \(u=(u^{2}_{i})>0\), \(v=(v^{2}_{i})>0\) such that
Without loss of generality, assume that \(u_{i}, v_{i}\in\mathbb{R}_{++}\). Let \(z=(z_{i})\) with \(z_{i}= u_{i} v_{i}\in\mathbb{R}_{++}\) and \(\mathcal{U} = \mathcal{P} \star\mathcal{Q}\). By Cauchy–Schwartz inequality, for \(1\leq i\leq n\), we have
It follows from the definitions of \(\alpha_{i}\), \(\beta_{i}\) and (18) that
Furthermore, using Lemma 3 and (19), one has
Case 2. Either \(\mathcal{P}\) or \(\mathcal{Q}\) is weakly reducible. Similar to the proof of Theorem 1, we obtain the desired result. □
In what follows, we give inclusion relations between Theorems 1 and 3.
Corollary 1
Let \(\mathcal{P}\) and \(\mathcal{Q}\) be strong M-tensors of order m and dimension n.
If \(p_{i\ldots i}q_{i\ldots i}\rho(J_{\mathcal{P}})\rho(J_{\mathcal {Q}})\leq\alpha_{i} \beta_{i}\) for \(i\in N\), then
if \(p_{i\ldots i}q_{i\ldots i}\rho(J_{\mathcal{P}})\rho(J_{\mathcal {Q}})\geq\alpha_{i} \beta_{i}\) for \(i\in N\), then
Proof
Observe that
When \(p_{i\ldots i}q_{i\ldots i}\rho(J_{\mathcal{P}})\rho(J_{\mathcal {Q}})\leq\alpha_{i} \beta_{i}\), from (22), we see
which implies
So, (20) holds.
If \(p_{i\ldots i}q_{i\ldots i}\rho(J_{\mathcal{P}})\rho(J_{\mathcal {Q}})\geq\alpha_{i} \beta_{i}\) for \(i\in N\), similar to the proof of (20), we obtain (21). □
Remark 2
If \(p_{i\ldots i}q_{i\ldots i}\rho(J_{\mathcal{P}})\rho(J_{\mathcal {Q}})\leq\alpha_{i} \beta_{i}\) for all \(1\leq i\leq n\), from (20), we verify that the bound of Theorem 1 is sharper than that of Theorem 3. When \(p_{i\ldots i}q_{i\ldots i}\rho(J_{\mathcal{P}})\rho(J_{\mathcal {Q}})\geq\alpha_{i} \beta_{i}\) for \(i\in N\), from (21), we deduce that the bound of Theorem 3 is tighter than that of Theorem 1.
The following examples give numerical comparisons between Theorems 1 and 3.
Example 3
Let \(\mathcal{P}=(p_{ijk})\), \(\mathcal{Q}=(q_{ijk})\) be defined in Example 2.
It is clear that \(\min_{1\leq i\leq n}(p_{i\ldots i}q_{i\ldots i})=9\). By computations, we get
Obviously, \(p_{i\ldots i}q_{i\ldots i}\rho(J_{\mathcal{P}})\rho (J_{\mathcal{Q}})\geq\alpha_{i} \beta_{i}\) for \(i=1,2,3\). From Theorem 1, we have
From Theorem 3, we have
So, the bound of Theorem 3 is tighter than that of Theorem 1.
Example 4
Let \(\mathcal{P}=(p_{ijk})\), \(\mathcal{Q}=(q_{ijk})\) be two tensors of order 3 and dimension 3 with elements defined as follows:
where
By computations, we get
From Theorem 1, one has
According to Theorem 3, we obtain
Thus, the bound of Theorem 1 is tighter than that of Theorem 3.
4 Conclusions
In this paper, we generalized important inequalities on the minimum eigenvalue for the Fan product from matrices to tensors. Based on characterizations of M-tensors, we proposed lower bound estimates on the minimum eigenvalue for the Fan product of two M-tensors. Finally, we gave some sufficient conditions to establish when particular inclusion relations hold.
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The authors are very indebted to the reviewers for their valuable comments and corrections, which improved the original manuscript of this paper.
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This research is supported by the Natural Science Foundation of China (11671228) and the Natural Science Foundation of Shandong Province (ZR2016AM10).
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Wang, G., Wang, Y. & Zhang, Y. Some inequalities for the Fan product of M-tensors. J Inequal Appl 2018, 257 (2018). https://doi.org/10.1186/s13660-018-1853-1
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DOI: https://doi.org/10.1186/s13660-018-1853-1