Sharp Bounds on the Minimum M-Eigenvalue of Elasticity M-Tensors

Abstract

The M-eigenvalue of elasticity M-tensors play important roles in nonlinear elastic material analysis. In this paper, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors without irreducible conditions, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.

1. Introduction

Tensor eigenvalue problems play an important role in numerical multilinear algebra [1,2,3,4,5,6,7], and they have a wide range in medical resonance [8], imaging spectral hypergraph theory [9], automatical control [10,11,12,13]. Particularly, the eigenvalue problem of the fourth-order elastic modulus tensor was dealt with by Love for the isotropic tensor [14] and for the anisotropic tensor [15,16,17,18,19,20,21]. A fourth-order real tensor $\mathcal{A}=(a_{ijkl})$ is called a partially symmetric tensor, denoted by $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n},$ if

$$a_{ijkl}=a_{jikl}=a_{ijlk},i,j,k,l\in N=\{1,2,\dots ,n\}.$$

(1)

A fourth-order partially symmetric tensor is useful in nonlinear elastic material analysis [3,16,17,18,19,20,21,22,23,24,25,26,27]. Ostrosablin [16] first constructed a complete system of eigentensors for the fourth rank tensor of elastic modulus, and Nikabadze [18] generalized these results and constructed a full system of eigentensors for a tensor of any even rank, as well as a complete system of eigentensor-columns for a tensor-block matrix of any even rank [22,23]. For example, a fourth-order partially symmetric tensor with $n=2$ or $3,$ called the elasticity tensor, can be used in the two/three-dimensional field equations for a homogeneous compressible nonlinearly elastic material for static problems without body forces [27]. To identify the strong ellipticity in elastic mechanics, Han et al. [25] introduced M-eigenvalues of a fourth-order partially symmetric tensor. For $\lambda \in \mathbb{R},x,y\in {\mathbb{R}}^n$, if

$$\left\{\begin{array}{c}\mathcal{A}·x{y}^2=\lambda x\hfill \\ \mathcal{A}{x}^{2}y·=\lambda y\hfill \\ x^{\top }x=1\hfill \\ y^{\top }y=1,\hfill \end{array}\right.$$

(2)

where ${(\mathcal{A}·x{y}^2)}_i={\displaystyle \sum _{j,k,l\in \left[n\right]}}{a}_{ijkl}{x}_j{y}_k{y}_l$, ${(\mathcal{A}{x}^{2}y·)}_l={\displaystyle \sum _{i,j,k\in \left[n\right]}}{a}_{ijkl}{x}_i{x}_j{y}_k$, then the scalar $\lambda$ is called an M-eigenvalue of the tensor $\mathcal{A}$, and x and y are called left and right M-eigenvectors of $\mathcal{A}$ associated with the M-eigenvalue. Then the M-spectral radius of $\mathcal{A}$ is denoted by

$${\rho }_M(\mathcal{A})=max\{|\lambda |:\lambda \in {\sigma }_M(\mathcal{A})\}.$$

Recently, tensors with special structures, such as nonnegative tensors, M-tensors and H-tensors, are becoming the focus in recent research [2,24,28,29,30,31]. Some effective algorithms for finding eigenvalue and the corresponding eigenvector have been implemented [1,24,32,33,34,35]. For example, Bozorgmanesh et al. [32] propose an algorithm that can solve E-eigenvalue problem faster. However, it is very difficult for these algorithms to compute all M-eigenvalues or E-eigenvalues. Thus, some researchers turned to investigating eigenvalue inclusion sets [4,7,36,37,38,39,40,41]. Particularly, some bounds for the minimum H-eigenvalue of nonsingular M-tensors have been proposed [2,28,30,42,43]. Ding et al. [24] introduced a structured partially symmetric tensor named elasticity M-tensors and established important properties of elasticity M-tensors and nonsingular elasticity M-tensors.

Definition 1.

$\mathcal{A}\in {\mathbb{E}}_{4,n}$ is called an elasticity M-tensor if there exist a nonnegative tensor $\mathcal{B}\in {\mathbb{E}}_{4,n}$ and a real number $s\ge {\rho }_M(\mathcal{B})$ such that

$$\mathcal{A}=s{\mathcal{I}}_M-\mathcal{B},$$

where ${\rho }_M(\mathcal{B})$ is the M-spectral radius and ${\mathcal{I}}_M=(e_{ijkl})\in {\mathbb{E}}_{4,n}$ is called elasticity identity tensor with its entries

$$e_{ijkl}=\left\{\begin{array}{cc}1,\hfill & ifi=jandk=l\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.$$

Furthermore, if $s>\rho (\mathcal{B}),$ then we call $\mathcal{A}$ a nonsingular elasticity M-tensor.

Based on structural properties of elasticity M-tensors, He et al. [26] proposed some bounds for the minimum M-eigenvalue under irreducible conditions. However, some of information eigenvectors x on elasticity M-tensors is not fully mined, such as $\underset{i,j\in N,i\ne j}{max}|x_i||x_j|\le \frac{1}{2}.$ Meanwhile, irreducibility is a relatively strict condition for elasticity M-tensor. Inspired by these observations, we want to present sharp bounds for the minimum M-eigenvalue of elasticity M-tensors by describing eigenvectors precisely without irreducible conditions, which improve existing results in [26].

This paper is organized as follows. In Section 2, some preliminary results are recalled. In Section 3, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors. Numerical examples are proposed to verify the efficiency of the obtained results.

2. Preliminaries

In this section, we firstly introduce some definitions and important properties of elasticity M-tensors [24,26,27].

Definition 2.

Let $\mathcal{A}=(a_{i_1{i}_2\dots i_m})\in {\mathbb{R}}^{[m,n]}$ be a square tensor, then $\mathcal{A}=(a_{i_1{i}_2\dots i_m})$ is called reducible if there exists a nonempty proper index subset $J\subset \{1,2,\dots ,n\}$ such that $a_{i_1{i}_2\dots i_m}=0,$ $\forall i_1\in J,\forall i_2,\dots ,i_m\notin J.$ If $\mathcal{A}$ is not reducible, then $\mathcal{A}$ is irreducible.

Lemma 1

(Theorem 1 of [27]). M-eigenvalues always exist. If x and y are left and right M-eigenvectors of $\mathcal{A}$, associated with an M-eigenvalue λ, then then $\lambda =\mathcal{A}{x}^2{y}^2.$

Lemma 2

(Lemma 2.3 of [26]). Let $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n}$ be an irreducible elasticity tensor and ${\tau }_M(\mathcal{A})$ be the minimal M-eigenvalues of $\mathcal{A}$, then

$${\tau }_M(\mathcal{A})\le \underset{i,l\in N}{min}\{a_{iill}\}.$$

Lemma 3

(Lemma 2.3 of [26]). Let $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n}$ be an irreducible and elasticity M-tensors and ${\tau }_M(\mathcal{A})$ be the minimal M-eigenvalue of $\mathcal{A}$. Then ${\tau }_M(\mathcal{A})\ge 0$ is an M-eigenvalue of $\mathcal{A}$ with positive eigenvectors.

Lemma 4

(Theorem 4.1 of [24]). The M-spectral radius of any nonnegative tensor in ${\mathbb{E}}_{4,n}$ is exactly its greatest M-eigenvalue. Furthermore, there is a pair of nonnegative M-eigenvectors corresponding to the M-spectral radius.

In the following, we characterize M-eigenvectors of elasticity M-tensors without irreducibility conditions.

Lemma 5.

Let $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n}$ be a elasticity M-tensor and ${\tau }_M(\mathcal{A})$ be the minimal M-eigenvalue. Then, there is a nonnegative M-eigenvector corresponding to ${\tau }_M(\mathcal{A})\ge 0$.

Proof. 

Since $\mathcal{A}$ is a elasticity M-tensor, there exist a nonnegative tensor $\mathcal{B}\in {\mathbb{E}}_{4,n}$ and a real number $c\ge {\rho }_M(\mathcal{B})$ such that

$$\mathcal{A}=c{\mathcal{I}}_M-\mathcal{B},$$

where ${\rho }_M(\mathcal{B})$ is the greatest M-eigenvalue of $\mathcal{B}$ with nonnegative eigenvectors by Theorem 4.1 in [24]. Setting ${\tau }_M(\mathcal{A})=c-{\rho }_M(\mathcal{B}),$ we have ${\tau }_M(\mathcal{A})\ge 0.$ It follows from Proposition 2.2 in [24] that ${\tau }_M(\mathcal{A})$ and ${\rho }_M(\mathcal{B})$ have the same eigenvectors. It follows from Lemma 4 that there exists a nonnegative M-eigenvector corresponding to ${\tau }_M(\mathcal{A})\ge 0$. Thus, the conclusion follows. ☐

3. Bounds for the Minimum $\mathbf{M}$-Eigenvalue of Elasticity $\mathbf{M}$-Tensors

In this section, we establish sharp bounds for ${\tau }_M(\mathcal{A}).$ We begin our work by collecting the information of $\underset{i,j\in N,i\ne j}{max}{x}_i^2{x}_j^2.$

Lemma 6.

For any $x\in {\mathbb{R}}^n,$ if

$$x_1^2+x_2^2+\cdots +x_n^2=1,$$

then

$$\underset{i,j\in N,i\ne j}{max}{x}_i^2{x}_j^2\le \frac{1}{4}.$$

Further, $\underset{i,j\in N,i\ne j}{max}{x}_i{x}_j\le \frac{1}{2}$ for all $x_i,x_j\ge 0.$

Proof. 

Define

$$f(x_1,\cdots ,x_n)=x_i^2{x}_j^2-\lambda (x_1^2+x_2^2+\cdots +x_n^2-1),$$

where $\lambda$ denotes Lagrange multiplier. For all $i\ne j$, deriving the above equation $x_i$ and $x_j$ respectively, we get

$$\left\{\begin{array}{c}2x_j^2{x}_i=2\lambda x_i,\hfill \\ 2x_i^2{x}_j=2\lambda x_j.\hfill \end{array}\right.$$

Hence, we obtain $x_i^2=x_j^2,$ $\lambda =\frac{1}{2}.$ Particularly, set

$$x_i=\pm x_j=\pm \frac{\sqrt{2}}{2},x_n=0,n\ne i,j$$

with $x_1^2+x_2^2+\cdots +x_n^2=1.$ So,

$$\underset{i,j\in N,i\ne j}{max}{x}_i^2{x}_j^2\le \frac{1}{4}.$$

Further,

$$\underset{i,j\in N,i\ne j}{max}{x}_i{x}_j\le \frac{1}{2},\forall x_i,x_j\ge 0.$$

 ☐

Remark 1.

For the right M-eigenvector $y\in {\mathbb{R}}^n,$ we can establish similar conclusions that $\underset{i,j\in N,i\ne j}{max}{y}_i{y}_j\le \frac{1}{2}$ for all $y_i,y_j\ge 0.$

Without irreducible conditions, we propose a sharp upper bound for the minimum M-eigenvalue of elasticity M-tensors.

Theorem 1.

Let $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n}$ be a elasticity M-tensor. Then,

$${\tau }_M(\mathcal{A})\le min\{\underset{i,l\in N}{min}{a}_{iill},\frac{{\displaystyle \sum _{i\in N}}{S}_i(\mathcal{A})}{n^2}\},$$

where $S_i(\mathcal{A})={\displaystyle \sum _{j,k,l\in N}}{a}_{ijkl}.$

Proof. 

Let ${\tau }_M(\mathcal{A})$ be the minimum M-eigenvalue of $\mathcal{A}$. It follows Lemma 1 that

$${\tau }_M(\mathcal{A})=\underset{x,y}{min}\{f_{\mathcal{A}}(x,y)=\mathcal{A}{x}^2{y}^2:x^{\top }x=1and{y}^{\top }y=1\}.$$

(3)

Setting a feasible solution of (3)

$$(\overline{x},\overline{y})=(\frac{1}{\sqrt{n}},\dots ,\frac{1}{\sqrt{n}},\frac{1}{\sqrt{n}},\dots ,\frac{1}{\sqrt{n}}),$$

we obtain

$${\tau }_M(\mathcal{A})\le f_{\mathcal{A}}(\overline{x},\overline{y})=\sum _{i,j\in N}\sum _{k,l\in N}\frac{a_{ijkl}}{n^2}=\frac{{\displaystyle \sum _{i\in \left[n\right]}}{S}_i(\mathcal{A})}{n^2}.$$

(4)

From Lemma 2 and (4), it holds that

$${\tau }_M(\mathcal{A})\le min\{{\displaystyle \underset{i,l\in N}{min}}{a}_{iill},\frac{\sum _{i\in N}{S}_i(\mathcal{A})}{n^2}\}.$$

 ☐

Next, we propose sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors.

Theorem 2.

Let $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n}$ be a elasticity M-tensor. Then

$${\tau }_M(\mathcal{A})\ge max\{\underset{i\in N}{min}\{{\alpha }_i-G_i(\mathcal{A})\},\underset{l\in N}{min}\{{\beta }_l-M_l(\mathcal{A})\}\}$$

where

$$G_i(\mathcal{A})={\omega }_i(\mathcal{A})-\frac{1}{2}{r}_i(\mathcal{A}),{\alpha }_i=\underset{l\in N}{min}\{a_{iill}\},{\omega }_i(\mathcal{A})=\underset{l\in N}{max}({\alpha }_i-a_{iill}-\sum _{\begin{array}{c}j=1,\\ j\ne i\end{array}}^n{a}_{ijll}),r_i(\mathcal{A})=\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{ijkl},$$

$$M_l(\mathcal{A})=m_l(\mathcal{A})-\frac{1}{2}{c}_l(\mathcal{A}),{\beta }_l=\underset{i\in N}{min}\{a_{iill}\},m_l(\mathcal{A})=\underset{i\in N}{max}({\beta }_l-a_{iill}-\sum _{\begin{array}{c}k=1,\\ k\ne l\end{array}}^n{a}_{iikl}),c_l(\mathcal{A})=\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkl}.$$

Proof. 

Let ${\tau }_M(\mathcal{A})$ be the minimum M-eigenvalue of $\mathcal{A}.$ By Lemma 5, there exist nonnegative left and right M-eigenvectors $(x,y)$ corresponding to ${\tau }_M(\mathcal{A}).$ On one hand, setting $x_p=\underset{k\in N}{max}\{x_k\},$ by $x^{\top }x=1$, one has $0<|x_p|\le 1.$ Recalling the p-th equation of ${\tau }_M(\mathcal{A})x=\mathcal{A}x{y}^2,$ we obtain

$$((a_{pp11}{y}_1^2+\cdots +a_{ppnn}{y}_n^2)-\tau (\mathcal{A}))x_p=-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_j{y}_k{y}_l-\sum _{\begin{array}{c}j,l=1,\\ j\ne p\end{array}}^n{a}_{pjll}{x}_j{y}_l^2,$$

(5)

Setting ${\alpha }_p=\underset{l\in N}{min}\{a_{ppll}\},$ by (5) and Lemma 6, one has

$$\begin{array}{cc}\hfill ({\alpha }_p-{\tau }_M(\mathcal{A}))x_p& =\sum _{\begin{array}{c}l=1,\\ j=p\end{array}}^n({\alpha }_p-a_{ppll})y_l^2{x}_p-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_j{y}_k{y}_l-\sum _{\begin{array}{c}j,l=1,\\ j\ne p\end{array}}^n{a}_{pjll}{x}_j{y}_l^2\hfill \\ \hfill & \le \sum _{\begin{array}{c}l=1,\\ j=p\end{array}}^n({\alpha }_p-a_{ppll})y_l^2{x}_p-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_p{y}_k{y}_l-\sum _{\begin{array}{c}j,l=1,\\ j\ne p\end{array}}^n{a}_{pjll}{x}_p{y}_l^2\hfill \\ \hfill & =\sum _{l=1}^n({\alpha }_p-a_{ppll}-\sum _{\begin{array}{c}j=1,\\ j\ne p\end{array}}^n{a}_{pjll})y_l^2{x}_p-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_p{y}_k{y}_l\hfill \\ \hfill & \le \underset{l\in N}{max}({\alpha }_p-a_{ppll}-\sum _{\begin{array}{c}j=1,\\ j\ne p\end{array}}^n{a}_{pjll})x_p-\frac{1}{2}\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_p.\hfill \end{array}$$

(6)

It follows from (6) and definition of ${\omega }_p$ that

$$({\alpha }_p-{\tau }_M(\mathcal{A}))x_p\le ({\omega }_p(\mathcal{A})-\frac{1}{2}{r}_p(\mathcal{A}))x_p,$$

which implies

$${\tau }_M(\mathcal{A})\ge {\alpha }_p-{\omega }_p(\mathcal{A})+\frac{1}{2}{r}_p(\mathcal{A}).$$

(7)

On the other hand, setting $y_t=\underset{k\in N}{max}\{y_k\},$ from the t-th equation of ${\tau }_M(\mathcal{A})y=\mathcal{A}{x}^{2}y,$ we obtain

$$((a_{11tt}{y}_1^2+\cdots +a_{nntt}{y}_n^2)-\tau (\mathcal{A}))y_t=-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{x}_i{x}_j{y}_k-\sum _{\begin{array}{c}i,k=1,\\ k\ne t\end{array}}^n{a}_{iikt}{x}_i^2{y}_k.$$

(8)

Letting ${\beta }_t=\underset{i\in N}{min}\{a_{iitt}\},$ by (8) and Lemma 6, we have

$$\begin{array}{cc}\hfill ({\beta }_t-{\tau }_M(\mathcal{A}))y_t& =\sum _{\begin{array}{c}i=1,\\ k=t\end{array}}^n({\beta }_t-a_{iitt})x_i^2{y}_t-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{x}_i{x}_j{y}_k-\sum _{\begin{array}{c}i,k=1,\\ k\ne t\end{array}}^n{a}_{iikt}{x}_i^2{y}_k\hfill \\ \hfill & \le \sum _{\begin{array}{c}i=1,\\ k=t\end{array}}^n({\beta }_t-a_{iitt})x_i^2{y}_t-\sum _{\begin{array}{c}i,k=1,\\ k\ne t\end{array}}^n{a}_{iikt}{y}_t{x}_i^2-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{x}_i{x}_j{y}_t\hfill \\ \hfill & =\sum _{l=1}^n({\beta }_t-a_{iitt}-\sum _{\begin{array}{c}k=1,\\ k\ne t\end{array}}^n{a}_{iikt})x_i^2{y}_t-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{x}_i{x}_j{y}_t\hfill \\ \hfill & \le \underset{i\in N}{max}({\beta }_t-a_{iitt}-\sum _{\begin{array}{c}k=1,\\ k\ne t\end{array}}^n{a}_{iikt})y_t-\frac{1}{2}\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{y}_t.\hfill \end{array}$$

(9)

It follows from (9) and definition of $m_t$ that

$$({\beta }_t-{\tau }_M(\mathcal{A}))y_t\le (m_t(\mathcal{A})-\frac{1}{2}{c}_t(\mathcal{A}))y_t,$$

which shows

$${\tau }_M(\mathcal{A})\ge {\beta }_t-m_t(\mathcal{A})+\frac{1}{2}{c}_t(\mathcal{A}).$$

(10)

From (7) and (10), the result follows. ☐

Now, we are at a position to prove that the bound in Theorem 2 is tighter than that of Theorem 3.1 of [26].

Corollary 1.

Let $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n}$ be a elasticity M-tensor. Then

$$max\{\underset{i\in N}{min}\{{\alpha }_i-G_i(\mathcal{A})\},\underset{l\in N}{min}\{{\beta }_l-M_l(\mathcal{A})\}\}\ge max\{\underset{i\in N}{min}\{{\alpha }_i-R_i(\mathcal{A})\},\underset{l\in N}{min}\{{\beta }_l-C_l(\mathcal{A})\}\}.$$

Proof. 

On one hand, it follows from Theorem 3.1 of [26] that

$${\alpha }_i-R_i(\mathcal{A})={\alpha }_i-{\gamma }_i(\mathcal{A})-r_i(\mathcal{A})={\alpha }_i-\underset{l\in N}{max}\{\sum _{\begin{array}{c}j=1,\\ j\ne i\end{array}}^n|a_{ijll}|\}-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n\left|a_{ijkl}\right|.$$

Since ${\alpha }_i-a_{iill}\le 0$ and ${\displaystyle \sum _{\begin{array}{c}j=1,\\ j\ne i\end{array}}^n}{a}_{ijll}\le 0$, we can verify

$$\underset{l\in N}{max}\{\sum _{\begin{array}{c}j=1,\\ j\ne i\end{array}}^n|a_{ijll}|\}\ge \underset{l\in N}{max}({\alpha }_i-a_{iill}-\sum _{\begin{array}{c}j=1,\\ j\ne i\end{array}}^n{a}_{ijll}),\frac{1}{2}\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{ijkl}\ge -\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n\left|a_{ijkl}\right|,$$

which shows

$${\alpha }_i-G_i(\mathcal{A})\ge {\alpha }_i-R_i(\mathcal{A}),\forall i\in N.$$

(11)

On the other hand, it follows from Theorem 3.1 of [26] that

$${\beta }_l-C_l(\mathcal{A})={\beta }_l-{\delta }_l(\mathcal{A})-\frac{1}{2}{c}_l(\mathcal{A})={\beta }_l-\underset{i\in N}{max}({\beta }_l-a_{iill}-\sum _{\begin{array}{c}k=1,\\ k\ne l\end{array}}^n{a}_{ijll})+\frac{1}{2}\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkl}.$$

Following the similar arguments in the proof of (11), we obtain

$${\beta }_l-{\mathcal{M}}_l(\mathcal{A})\ge {\beta }_l-{\mathcal{C}}_l(\mathcal{A}),\forall l\in N.$$

(12)

It follows from (11) and (12) that

$$max\{\underset{i\in N}{min}\{{\alpha }_i-G_i(\mathcal{A})\},\underset{l\in N}{min}\{{\beta }_l-M_l(\mathcal{A})\}\}\ge max\{\underset{i\in N}{min}\{{\alpha }_i-R_i(\mathcal{A})\},\underset{l\in N}{min}\{{\beta }_l-C_l(\mathcal{A})\}\}.$$

 ☐

Choosing $x_q$ as a component of x with the second largest modulus, we obtain another sharp lower bound for ${\tau }_M(\mathcal{A}).$

Theorem 3.

Let $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n}$ be a elasticity M-tensor. Then

$${\tau }_M(\mathcal{A})\ge max\{{\varphi }_1(\mathcal{A}),{\varphi }_2(\mathcal{A})\},$$

where$\begin{array}{c}{\varphi }_1(\mathcal{A})=\underset{i,v\in N,v\ne i}{min}\{\frac{1}{2}\{{\alpha }_i+\frac{1}{2}{r}_i(\mathcal{A})+{\alpha }_v-{\omega }_v^i(\mathcal{A})-{\Delta }_{i,v}^{\frac{1}{2}}(\mathcal{A})\},{\alpha }_i+\frac{1}{2}{r}_i(\mathcal{A}),{\alpha }_v-{\omega }_v^i(\mathcal{A})\},\hfill \\ {\varphi }_2(\mathcal{A})=\underset{u,l\in N,u\ne l}{min}\{\frac{1}{2}\{{\beta }_l+\frac{1}{2}{c}_l(\mathcal{A})+{\beta }_u-m_u^l(\mathcal{A})-{\theta }_{l,u}^{\frac{1}{2}}(\mathcal{A})\},{\beta }_l+\frac{1}{2}{c}_l(\mathcal{A}),{\beta }_u-m_u^l(\mathcal{A})\},\hfill \\ {\Delta }_{i,v}(\mathcal{A})={({\alpha }_i+\frac{1}{2}{r}_i(\mathcal{A})-{\alpha }_v+{\omega }_v^i(\mathcal{A}))}^2+4{\omega }_i(\mathcal{A})({\gamma }_v^i(\mathcal{A})-\frac{1}{2}{r}_v(\mathcal{A})),\hfill \\ {\theta }_{l,u}(\mathcal{A})={({\beta }_l+\frac{1}{2}{c}_l(\mathcal{A})-{\beta }_u+m_u^l(\mathcal{A}))}^2+4m_l(\mathcal{A})({\delta }_u^l(\mathcal{A})-\frac{1}{2}{c}_u(\mathcal{A})),\hfill \\ {\gamma }_v^i(\mathcal{A})=\underset{l\in N}{max}(-a_{vill}),{\omega }_v^i(\mathcal{A})=\underset{l\in N}{max}({\alpha }_v-a_{vvll}-{\displaystyle \sum _{j=1;j\ne v,i}^n}{a}_{vjll}),\hfill \\ {\delta }_u^l(\mathcal{A})=\underset{i\in N}{max}(-a_{iilu}),m_u^l(\mathcal{A})=\underset{i\in N}{max}({\beta }_u-a_{iiuu}-{\displaystyle \sum _{k=1;k\ne u,l}^n}{a}_{iikl}).\hfill \end{array}$

Proof. 

Let ${\tau }_M(\mathcal{A})$ be the minimal M-eigenvalue of $\mathcal{A}.$ By Lemma 5, there exist nonnegative left and right M-eigenvectors $(x,y)$ corresponding to ${\tau }_M(\mathcal{A}).$ On one hand, set $x_p\ge x_q\ge \underset{k\in N,k\ne p,q}{max}\{x_k\}.$ By the p-th equation of ${\tau }_M(\mathcal{A})x=\mathcal{A}x{y}^2,$ one has

$$((a_{pp11}{y}_1^2+\cdots +a_{ppnn}{y}_n^2)-{\tau }_M(\mathcal{A}))x_p=-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_j{y}_k{y}_l-\sum _{\begin{array}{c}j,l=1,\\ j\ne p\end{array}}^n{a}_{pjll}{x}_j{y}_l^2.$$

(13)

Setting ${\alpha }_p=\underset{l\in N}{min}\{a_{ppll}\},$ from (13) and Lemma 6, we obtain

$$\begin{array}{cc}\hfill ({\alpha }_p-{\tau }_M(\mathcal{A}))x_p& =\sum _{\begin{array}{c}l=1,\\ j=p\end{array}}^n({\alpha }_p-a_{ppll})y_l^2{x}_p-\sum _{\begin{array}{c}j,l=1,\\ j\ne p\end{array}}^n{a}_{pjll}{x}_j{y}_l^2-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_j{y}_k{y}_l\hfill \\ \hfill & \le \sum _{\begin{array}{c}l=1,\\ j=p\end{array}}^n({\alpha }_p-a_{ppll})y_l^2{x}_q-\sum _{\begin{array}{c}j,l=1,\\ j\ne p\end{array}}^n{a}_{pjll}{x}_q{y}_l^2-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_p{y}_k{y}_l\hfill \\ \hfill & =\sum _{l=1}^n({\alpha }_p-a_{ppll}-\sum _{\begin{array}{c}j=1,\\ j\ne p\end{array}}^n{a}_{pjll})y_l^2{x}_q-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_p{y}_k{y}_l\hfill \\ \hfill & \le \underset{l\in N}{max}({\alpha }_p-a_{ppll}-\sum _{\begin{array}{c}j=1,\\ j\ne p\end{array}}^n{a}_{pjll})x_q-\frac{1}{2}\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{pjkl}{x}_p,\hfill \end{array}$$

which implies

$$({\alpha }_p-{\tau }_M(\mathcal{A})+\frac{1}{2}{r}_p(\mathcal{A}))x_p\le {\omega }_p(\mathcal{A})x_q.$$

(14)

From the q-th equation of ${\tau }_M(\mathcal{A})x=\mathcal{A}x{y}^2$ and ${\alpha }_q=\underset{l\in N}{min}\{a_{qqll}\},$ we yield

$$\begin{array}{cc}\hfill & ({\alpha }_q-{\tau }_M(\mathcal{A}))x_q=\sum _{\begin{array}{c}l=1,\\ j=q\end{array}}^n({\alpha }_q-a_{qqll})y_l^2{x}_q-\sum _{\begin{array}{c}j,l=1,\\ j\ne q\end{array}}^n{a}_{qjll}{x}_j{y}_l^2-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{qjkl}{x}_j{y}_k{y}_l\hfill \\ \hfill & \le \sum _{\begin{array}{c}l=1,\\ j=q\end{array}}^n({\alpha }_q-a_{qqll})y_l^2{x}_q-\sum _{\begin{array}{c}j,l=1,\\ j\ne q,p\end{array}}^n{a}_{qjll}{x}_q{y}_l^2-\sum _{\begin{array}{c}l=1,\\ j=p\end{array}}^n{a}_{qpll}{x}_p{y}_l^2-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{qjkl}{x}_p{y}_k{y}_l\hfill \\ \hfill & =\sum _{l=1}^n({\alpha }_q-a_{qqll}-\sum _{\begin{array}{c}j=1,\\ j\ne q,p\end{array}}^n{a}_{qjll})y_l^2{x}_q-\sum _{\begin{array}{c}l=1,\\ j=p\end{array}}^n{a}_{qpll}{x}_p{y}_l^2-\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{qjkl}{x}_p{y}_k{y}_l\hfill \\ \hfill & \le \underset{l\in N}{max}({\alpha }_q-a_{qqll}-\sum _{\begin{array}{c}j=1,\\ j\ne q,p\end{array}}^n{a}_{qjll})x_q+\underset{l\in N}{max}(-a_{qpll})x_p-\frac{1}{2}\sum _{\begin{array}{c}j,k,l=1,\\ k\ne l\end{array}}^n{a}_{qjkl}{x}_p,\hfill \end{array}$$

(15)

which implies

$$({\alpha }_q-{\tau }_M(\mathcal{A})-{\omega }_q^p(\mathcal{A}))x_q\le ({\gamma }_q^p(\mathcal{A})-\frac{1}{2}{r}_q(\mathcal{A}))x_p.$$

(16)

When ${\alpha }_p-{\tau }_M(\mathcal{A})+\frac{1}{2}{r}_p(\mathcal{A})\ge 0$ or ${\alpha }_q-{\tau }_M(\mathcal{A})-{\omega }_q^p(\mathcal{A})\ge 0,$ multiplying inequalities (14) with (16), one has

$$({\alpha }_p-{\tau }_M(\mathcal{A})+\frac{1}{2}{r}_p(\mathcal{A}))({\alpha }_q-{\tau }_M(\mathcal{A})-{\omega }_q^p(\mathcal{A}))\le {\omega }_p(\mathcal{A})({\gamma }_q^p(\mathcal{A})-\frac{1}{2}{r}_q(\mathcal{A})).$$

Then, solving for ${\tau }_M(\mathcal{A}),$ we have

$${\tau }_M(\mathcal{A})\ge \frac{1}{2}({\alpha }_p+\frac{1}{2}{r}_p(\mathcal{A})+{\alpha }_q-{\omega }_q^p(\mathcal{A})-{\Delta }_{p,q}^{\frac{1}{2}}(\mathcal{A})),$$

(17)

where ${\Delta }_{p,q}(\mathcal{A})={({\alpha }_p+\frac{1}{2}{r}_p(\mathcal{A})-{\alpha }_q+{\omega }_q^p(\mathcal{A}))}^2+4{\omega }_p(\mathcal{A})({\gamma }_q^p(\mathcal{A})-\frac{1}{2}{r}_q(\mathcal{A})).$

When ${\alpha }_p-{\tau }_M(\mathcal{A})+\frac{1}{2}{r}_p(\mathcal{A})<0$ and ${\alpha }_q-{\tau }_M(\mathcal{A})-{\omega }_q^p(\mathcal{A})<0,$ one has

$${\tau }_M(\mathcal{A})>{\alpha }_p+\frac{1}{2}{r}_p(\mathcal{A})\mathrm{and}{\tau }_M(\mathcal{A})>{\alpha }_q-{\omega }_q^p(\mathcal{A}).$$

(18)

It follows from (17) and (18) that

$${\tau }_M(\mathcal{A})\ge \underset{p,q\in N,q\ne p}{min}\{\frac{1}{2}\{{\alpha }_p+\frac{1}{2}{r}_p(\mathcal{A})+{\alpha }_q-{\omega }_q^p(\mathcal{A})-{\Delta }_{p,q}^{\frac{1}{2}}(\mathcal{A})\},{\alpha }_p+\frac{1}{2}{r}_p(\mathcal{A}),{\alpha }_q-{\omega }_q^p(\mathcal{A})\}.$$

On the other hand, set $y_t\ge y_s\ge \underset{k\in N,k\ne t}{max}\{y_k\}.$ From the t-th equation of ${\tau }_M(\mathcal{A})y=\mathcal{A}{x}^{2}y,$ it follows that

$$((a_{11tt}{x}_1^2+\cdots +a_{nntt}{x}_n^2)-{\tau }_M(\mathcal{A}))y_t=-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{x}_i{x}_j{y}_k-\sum _{\begin{array}{c}i,k=1,\\ k\ne t\end{array}}^n{a}_{iikt}{x}_i^2{y}_k.$$

(19)

Letting ${\beta }_t=\underset{i\in N}{min}\{a_{iitt}\},$ by (19) and Lemma 6, we obtain

$$\begin{array}{cc}\hfill & ({\beta }_t-{\tau }_M(\mathcal{A}))y_t=\sum _{\begin{array}{c}i=1,\\ k=t\end{array}}^n({\beta }_t-a_{iitt})x_i^2{y}_t-\sum _{\begin{array}{c}i,k=1,\\ k\ne t\end{array}}^n{a}_{iikt}{x}_i^2{y}_k-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{x}_i{x}_j{y}_k\hfill \\ \hfill & \le \sum _{\begin{array}{c}i=1,\\ k=t\end{array}}^n({\beta }_t-a_{iitt})x_i^2{y}_s-\sum _{\begin{array}{c}i,k=1,\\ k\ne t\end{array}}^n{a}_{iikt}{x}_i^2{y}_s-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{x}_i{x}_j{y}_t\hfill \\ \hfill & =\sum _{i=1}^n({\beta }_t-a_{iitt}-\sum _{\begin{array}{c}k=1,\\ k\ne t\end{array}}^n{a}_{iikt})x_i^2{y}_s-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{x}_i{x}_j{y}_t\hfill \\ \hfill & \le \underset{i\in N}{max}({\beta }_t-a_{iitt}-\sum _{\begin{array}{c}k=1,\\ k\ne t\end{array}}^n{a}_{iikt})y_s-\frac{1}{2}\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijkt}{y}_t.\hfill \end{array}$$

(20)

Using (20), we yield

$$({\beta }_t-{\tau }_M(\mathcal{A})+\frac{1}{2}{c}_t(\mathcal{A}))y_t\le m_t(\mathcal{A})y_s.$$

(21)

Recalling s-th equation of $\tau (\mathcal{A})y=\mathcal{A}{x}^{2}y$ and ${\beta }_s=\underset{i\in N}{min}\{a_{iiss}\},$ we have

$$\begin{array}{cc}\hfill & ({\beta }_s-{\tau }_M(\mathcal{A}))y_s=\sum _{\begin{array}{c}i=1,\\ k=s\end{array}}^n({\beta }_s-a_{iiss})x_i^2{y}_s-\sum _{\begin{array}{c}i,k=1,\\ k\ne s\end{array}}^n{a}_{iiks}{x}_i^2{y}_k-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijks}{x}_i{x}_j{y}_k\hfill \\ \hfill & \le \sum _{\begin{array}{c}i=1,\\ k=s\end{array}}^n({\beta }_s-a_{iiss})x_i^2{y}_t-\sum _{\begin{array}{c}i,k=1,\\ k\ne s,t\end{array}}^n{a}_{iiks}{x}_i^2{y}_s-\sum _{\begin{array}{c}i=1,\\ k=t\end{array}}^2{a}_{iits}{x}_i^2{y}_t-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijks}{x}_i{x}_j{y}_t\hfill \\ \hfill & =\sum _{i=1}^n({\beta }_s-a_{iiss}-\sum _{\begin{array}{c}k=1,\\ k\ne s,t\end{array}}^n{a}_{iiks})x_i^2{y}_s-\sum _{\begin{array}{c}i=1,\\ k=t\end{array}}^n{a}_{iits}{x}_i^2{y}_t-\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijks}{x}_i{x}_j{y}_t\hfill \\ \hfill & \le \underset{i\in N}{max}({\beta }_s-a_{iiss}-\sum _{\begin{array}{c}k=1,\\ k\ne s,t\end{array}}^n{a}_{iiks})y_t+\underset{i\in N}{max}(-a_{iits})y_t-\frac{1}{2}\sum _{\begin{array}{c}i,j,k=1,\\ i\ne j\end{array}}^n{a}_{ijks}{y}_t,\hfill \end{array}$$

which implies

$$({\beta }_s-{\tau }_M(\mathcal{A})-m_s^t(\mathcal{A}))y_s\le ({\delta }_s^t(\mathcal{A})-\frac{1}{2}{c}_s(\mathcal{A}))y_t.$$

(22)

When ${\beta }_t-{\tau }_M(\mathcal{A})+\frac{1}{2}{c}_t(\mathcal{A})\ge 0$ or ${\beta }_s-{\tau }_M(\mathcal{A})-m_s^t(\mathcal{A})\ge 0,$ multiplying inequalities (21) with (22), one has

$$({\beta }_t-{\tau }_M(\mathcal{A})+\frac{1}{2}{c}_t(\mathcal{A}))({\beta }_s-{\tau }_M(\mathcal{A})-m_s^t(\mathcal{A}))\le m_t(\mathcal{A})({\delta }_s^t(\mathcal{A})-\frac{1}{2}{c}_s(\mathcal{A})).$$

Then, solving for $\tau (\mathcal{A}),$ we obtain

$${\tau }_M(\mathcal{A})\ge \frac{1}{2}\{{\beta }_t+\frac{1}{2}{c}_t(\mathcal{A})+{\beta }_s-m_s^t(\mathcal{A})-{\theta }_{t,s}^{\frac{1}{2}}(\mathcal{A})\},$$

(23)

where ${\theta }_{t,s}(\mathcal{A})={({\beta }_t+\frac{1}{2}{c}_t(\mathcal{A})-{\beta }_s+m_s^t(\mathcal{A}))}^2+4m_t(\mathcal{A})({\delta }_s^t(\mathcal{A})-\frac{1}{2}{c}_s(\mathcal{A})).$

When ${\beta }_t-{\tau }_M(\mathcal{A})+\frac{1}{2}{c}_t(\mathcal{A})<0$ and ${\beta }_s-{\tau }_M(\mathcal{A})-m_s^t(\mathcal{A})<0,$ then we have

$${\tau }_M(\mathcal{A})>{\beta }_t+\frac{1}{2}{c}_t(\mathcal{A})and{\tau }_M(\mathcal{A})>{\beta }_s-m_s^t(\mathcal{A}).$$

(24)

From (23) and (24), it holds that

$${\tau }_M(\mathcal{A})\ge \underset{t,s\in N,s\ne t}{min}\{\frac{1}{2}\{{\beta }_t+\frac{1}{2}{c}_t(\mathcal{A})+{\beta }_s-m_s^t(\mathcal{A})-{\theta }_{t,s}^{\frac{1}{2}}(\mathcal{A})\},{\beta }_t+\frac{1}{2}{c}_t(\mathcal{A}),{\beta }_s-m_s^t(\mathcal{A})\}.$$

Thus, the desired result holds. ☐

In the following, we use Example 3.1 of [26] to show the superiority of our results.

Example 1.

Let $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,2}$ be an elasticity M-tensor, whose entries are

$$a_{ijkl}=\left\{\begin{array}{c}{a}_{1111}=a_{1122}=4.1,a_{2211}=a_{2222}=5,\hfill \\ a_{1112}=a_{1121}=-1,a_{2212}=a_{2221}=-1,\hfill \\ a_{2111}=a_{1211}=-1,a_{1222}=a_{2122}=-1,\hfill \\ a_{ijkl}=0,otherwise.\hfill \end{array}\right.$$

The bounds via different estimations given in the literature are shown in Table 1:

Table 1. Bound estimations of the minimum M-eigenvalue with different methods

References	Interval
Lemma 2 and Theorem 3.1 of [26]	$1.10\le {\tau }_M(\mathcal{A})\le 4.10$
Lemma 2 and Theorem 3.2 of [26]	$1.29\le {\tau }_M(\mathcal{A})\le 4.10$
Theorem 1 and Theorem 2	$2.10\le {\tau }_M(\mathcal{A})\le 2.55$
Theorem 1 and Theorem 3	$2.35\le {\tau }_M(\mathcal{A})\le 2.55$

By computations, we obtain that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are $({\tau }_M(\mathcal{A}),\overline{x},\overline{y})=(2.4534,(0.8398,0.5430),(0.7071,0.7071)).$ It is easy to see that the results given in Theorems 3.1–3.3 are sharper than some existing results [26]. It is noted that Theorems 2 and 3 have their own advantages. Theorem 3 can estimate the lower bound of the minimum M-eigenvalue more accurately, but the calculation of Theorem 2 is simpler.

Ding et al. [24] pointed out that a tensor is M-positive if and only if its smallest M-eigenvalue is positive. In the following, the results given in Theorems 2 and 3 can exactly check the positiveness of the elasticity M-tensor $\mathcal{A}$.

Example 2.

Consider the elasticity M-tensor $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,2}$ defined by

$$a_{ijkl}=\left\{\begin{array}{c}{a}_{1111}=3.1,a_{1122}=4.1,a_{2211}=5,a_{2222}=6,\hfill \\ a_{1112}=a_{1121}=-1,a_{2212}=a_{2221}=-1,\hfill \\ a_{2111}=a_{1211}=-1,a_{1222}=a_{2122}=-2,\hfill \\ a_{1221}=a_{1212}=a_{2121}=a_{2112}=-0.5.\hfill \end{array}\right.$$

The bounds via different estimations given in the literature are shown in Table 2.

Table 2. Bound estimations of the minimum M-eigenvalue and testing the M-positive definiteness

References	Interval
Lemma 2 and Theorem 3.1 of [26]	$-1.90\le {\tau }_M(\mathcal{A})\le 3.10$
Lemma 2 and Theorem 3.2 of [26]	$-1.45\le {\tau }_M(\mathcal{A})\le 3.10$
Theorems 1 and 2	$0.60\le {\tau }_M(\mathcal{A})\le 1.55$
Theorems 1 and 3	$0.77\le {\tau }_M(\mathcal{A})\le 1.55$

By computations, we obtain that the minimum M-eigenvalue and corresponding with left and right M-eigenvectors are $({\tau }_M(\mathcal{A}),\overline{x},\overline{y})=(1.3350,(0.8462,0.5329),(0.7190,0.6951)).$ From Theorems 2 and 3, we obtain $0.77\le {\tau }_M(\mathcal{A})\le 1.55$, which shows that $\mathcal{A}$ is M-positive definite. However, the existing results of [26] cannot identify the M-positiveness of $\mathcal{A}.$

For the medium-sized tensors, we show the validity of the estimations by our theorems.

Example 3.

All testing elasticity M-tensors $\mathcal{A}=(a_{ijkl})\in {\mathbb{E}}_{4,n}$ are generated as follows: $a_{iijj}=n^2+4i+4j$ and other elements are generated randomly in $[-0.5,0]$ by MATLAB R2014a, where n denotes variable dimension. For different dimensions elasticity M-tensors, the values presented in the table are the average values of 10 examples. The bounds via different estimations given in the literature are shown in Table 3.

Table 3. Comparison estimations of the minimum M-eigenvalue with random elasticity M-tensors

References	$\mathit{n}=10$ Bounds	$\mathit{n}=20$ Bounds	$\mathit{n}=30$ Bounds
Lemma 2.2 and Theorem 3.1 of [9]	[−129.8, 105]	[−1561.9, 408]	[−5660.2, 908]
Lemma 2.2 and Theorem 3.2 of [9]	[−102.4, 105]	[−1324.4, 408]	[−4975.5, 908]
Theorems 1 and 2	[−8.9, 102.6]	[−530.4, 383.9]	[−2173.7, 799.7]
Theorems 1 and 3	[3.8, 102.6]	[−419.3, 383.9]	[−1956.8, 799.7]

4. Conclusions

In this paper, we exactly characterized the information of eigenvectors without irreducible conditions. Further, we proposed a new upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors by establishing new eigenvalue inequality. Numerical examples were proposed to verify the efficiency of the obtained results.