A new Brauer-type Z-eigenvalue inclusion set for even-order tensors

Abstract

A new Brauer-type Z-eigenvalue inclusion set for an even-order real tensor is presented. It is proved that it is tighter than the existing inclusion sets. As an application, a sufficient condition for the positive definiteness of an even-order real symmetric tensor (also a homogeneous polynomial form) and asymptotically stability of time-invariant polynomial systems is given.

1 Introduction

Let m and n be two positive integers with $m\ge 2$ and $n\ge 2$, $[n]=\{1,2,\ldots ,n\}$, $\mathbb {C}$ (resp. $\mathbb {R}$) be the set of all complex (resp. real) numbers, ${\mathbb {R}}^{n}$ be the set of all n-dimensional real vectors, $\mathbb {R}^{[m,n]}$ be the set of all order m dimension n real tensors. Let $x=(x_1,x_2,\ldots ,x_n)^\top \in \mathbb {R}^n$. Let $\mathscr {A}=(a_{i_{1}i_2\ldots i_{m}})\in \mathbb {R}^{[m,n]}$, i.e.,

$$\begin{aligned} a_{i_{1}i_2\ldots i_{m}}\in {\mathbb {R}},\quad i_j\in [n],\quad j\in [m]. \end{aligned}$$

Let $\varPi _m$ be the permutation group of m indices. If for any $\pi \in \varPi _m$,

$$\begin{aligned} a_{i_1i_2\ldots i_m}=a_{i_{\pi (1)}i_{\pi (2)}\ldots i_{\pi (m)}} \end{aligned}$$

then $\mathscr {A}$ is called a symmetric tensor [13].

If there are $\lambda \in \mathbb {R}$ and $x=(x_1,x_2,\ldots ,x_n)^{\top }\in {\mathbb {R}}^{n}\setminus \{0\}$ such that

$$\begin{aligned} \mathscr {A}x^{m-1}=\lambda x \quad \text{ and }\quad x^\top x=1, \end{aligned}$$

where $\mathscr {A}x^{m-1}$ is an n-dimensional vector, whose i-th component is

$$\begin{aligned} \left( \mathscr {A}x^{m-1}\right) _i=\sum \limits _{i_2,\ldots ,i_m\in [n]} a_{ii_2\ldots i_m}x_{i_2}\ldots x_{i_m}, \end{aligned}$$

then $\lambda$ is called a Z-eigenvalue of $\mathscr {A}$ and x is called a Z-eigenvector associated with $\lambda$ [10, 13]. Let $\sigma (\mathscr {A})$ be the set of all Z-eigenvalues of $\mathscr {A}$.

The Z-identity tensor is introduced by the authors in [7, 8, 13]. A tensor $\mathscr {E}=(e_{i_1i_2\ldots i_m})\in \mathbb {R}^{[m,n]}$ with m even is called a Z-identity tensor if for any vector $x\in \mathbb {R}^n$,

$$\begin{aligned} \mathscr {E}x^{m-1}= x \quad \text{ and }\quad x^\top x=1. \end{aligned}$$

Note here that an even-order n dimension Z-identity tensor is not unique in general. For instance, the following two tensors are both Z-identity tensors:

Case I. ([8, Definition 2.1]): Let $\mathscr {E}_1=(e_{i_1i_2\ldots i_m})\in \mathbb {R}^{[m,n]},$ where

$$\begin{aligned} e_{i_1i_1i_2i_2\ldots i_ki_k}=1,\quad i_1,i_2,\ldots ,i_k\in [n], \quad \text{ and }\quad m=2k; \end{aligned}$$

Case II. ([7, Property 2.4]): Let $\mathscr {E}_2=(e_{i_1i_2\ldots i_m})\in \mathbb {R}^{[m,n]}$, where

$$\begin{aligned} e_{i_1\ldots i_m}=\frac{1}{m!}\sum _{\pi \in \varPi _m}\delta _{i_{\pi (1)}i_{\pi (2)}}\delta _{i_{\pi (3)}i_{\pi (4)}}\ldots \delta _{i_{\pi (m-1)}i_{\pi (m)}}, \end{aligned}$$

where $\delta$ is the standard Kronecker delta, i,e., $\delta _{ij}=1$ if $i=j$ and $\delta _{ij}=0$ if $i\ne j$.

For convenient applications, the Z-identity tensor $\mathscr {E}_2=(e_{ijkl})\in \mathbb {R}^{[4,n]}$ is listed as follows:

$$\begin{aligned} e_{ijkl} = {\left\{ \begin{array}{ll} 1, &{}\quad \text {if } i = j = k = l, \\ 1/3, &{}\quad \text {if } i = j \ne k = l, \\ 1/3, &{}\quad \text {if } i = k \ne j = l, \\ 1/3, &{}\quad \text {if } i = l \ne j = k, \\ 0, &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

An even-order m dimension n real symmetric tensor $\mathscr {A}$ defines an m-th degree homogeneous polynomial

$$\begin{aligned} f(x)=\mathscr {A}x^{m}=\sum _{{i_1,i_2,\ldots ,i_m}\in [n]}a_{i_{1}i_{2}\ldots i_{m}}x_{i_1}x_{i_2}\ldots x_{i_m}. \end{aligned}$$

(1)

If $f(x)>0$ for any $x\in \mathbb {R}^{n}\setminus \{0\}$, then we call that f(x) is positive definite. It is pointed out that f(x) is positive definite if and only if $\mathscr {A}$ is positive definite [13, 14]. On the other hand, if all Z-eigenvalues of the real symmetric tensor $\mathscr {A}$ with order even are positive, then $\mathscr {A}$ is positive definite and therefore f(x) is also positive definite. The positive definiteness of f(x) has extremely important applications in real life. As pointed out in some documents, it is widely used in spectral hypergraph theory [15, 16], automatic control [12] and the stability of nonlinear systems [1, 2].

For judging the positive definiteness of f(x), we must calculate all Z-eigenvalues of an even-order real symmetric tensor $\mathscr {A}$, or calculate the minimum Z-eigenvalue of $\mathscr {A}$. When all Z-eigenvalues of $\mathscr {A}$ are greater than 0, or the minimum Z-eigenvalue of $\mathscr {A}$ is greater than 0, we can judge that f(x) is positive definite. However, if m or n are very large, it is difficult to calculate all Z-eigenvalues of $\mathscr {A}$ and the minimum Z-eigenvalue of $\mathscr {A}$. In order to be able to solve this problem quickly, we can take a very normal and simple method: we only need to judge the signs of all Z-eigenvalues, but not to compute all Z-eigenvalues. In order to achieve this goal, one can construct a set which includes all Z-eigenvalues of $\mathscr {A}$. If this set is just in the right-half complex plane, then he can conclude that all Z-eigenvalues are positive, and consequently, $\mathscr {A}$ is positive definite. The related results are shown in [8, 19,20,21, 24, 30, 31].

Wang et al. [25] gave the following Z-eigenvalue inclusion set for tensors as follows:

Theorem 1

[25, Theorem 3.2] Let $\mathscr {A}=(a_{i_{1}\ldots i_{m}})\in {\mathbb {R}}^{[m,n]}$. Then

$$\begin{aligned} \sigma (\mathscr {A})\subseteq \mathscr {L}(\mathscr {A})=\bigcup \limits _{i\in [n]}\bigcap \limits _{j\in [n],j\ne i}\mathscr {L}_{i,j}(\mathscr {A}). \end{aligned}$$

where

$$\begin{aligned} \mathscr {L}_{i,j}(\mathscr {A})=\{z\in \mathbb {C}: \left( \mid z\mid -(R_i(\mathscr {A})-\mid a_{ij\ldots j}\mid )\right) \mid z\mid \le \mid a_{ij\ldots j}\mid R_{j}(\mathscr {A})\} \end{aligned}$$

and

$$\begin{aligned} R_i(\mathscr {A})=\sum \limits _{i_2,\ldots ,i_m\in [n]}\mid a_{ii_2\ldots i_m}\mid . \end{aligned}$$

From Theorem 1, we can easily see that $0\in \mathscr {L}(\mathscr {A})$. Therefore, this means that we cannot use the set $\mathscr {L}(\mathscr {A})$ to determine the positive definiteness of a real symmetric tensor $\mathscr {A}$ of even order. However, there are many such similar sets, which can be seen in detail [3,4,5,6, 9, 11, 17, 18, 23, 25,26,27,28,29]. Because 0 exists in these inclusion sets, we cannot use such inclusion sets to determine the positive definiteness of even-order real symmetric tensors.

In order to overcome this drawback, Li et al. [8] presented a Z-eigenvalue inclusion interval with n parameters for even-order real tensors as follows:

Theorem 2

[8, Theorem 2.2] Let $\mathscr {A}=(a_{i_{1}\ldots i_{m}})\in {\mathbb {R}}^{[m,n]}$ with m even. Then for any real vector $\alpha =(\alpha _1,\ldots ,\alpha _n)^{\top }\in \mathbb {R}^{n}$,

$$\begin{aligned} \sigma (\mathscr {A})\subseteq \mathscr {G}(\mathscr {A},\alpha ):=\bigcup _{i\in [n]} \Big (\mathscr {G}_i(\mathscr {A},\alpha ):=\{z\in \mathbb {R}:\mid z-\alpha _i\mid \le R_{i}(\mathscr {A},\alpha _i)\}\Big ), \end{aligned}$$

where

$$\begin{aligned} R_i(\mathscr {A},\alpha _i)= \sum \limits _{\begin{array}{c} i_2,\ldots ,i_m\in [n], \\ e_{ii_2\ldots i_m}\ne 0 \end{array}}|a_{ii_2\ldots i_m}-\alpha _i e_{ii_2\ldots i_m}| +\sum \limits _{\begin{array}{c} i_2,\ldots ,i_m\in [n], \\ e_{ii_2\ldots i_m} = 0 \end{array}}|a_{ii_2\ldots i_m}|. \end{aligned}$$

In order to be able to locate the Z-eigenvalues more accurately, Shen et al. [22] gave the following inclusion set.

Theorem 3

[22, Theorem 1] Let $\mathscr {A}=(a_{i_{1}\ldots i_{m}})\in {\mathbb {R}}^{[m,n]}$ and $\mathscr {E}\in {\mathbb {R}}^{[m,n]}$ be a Z-identity tensor. For any real vector $\alpha =(\alpha _1,\ldots ,\alpha _n)^\top \in \mathbb {R}^{n}$, then

$$\begin{aligned} \sigma (\mathscr {A})\subseteq \varUpsilon (\mathscr {A},\alpha )=\bigcup \limits _{i\in [n]}\bigcap \limits _{j\in [n],j\ne i}\varUpsilon _{i,j}(\mathscr {A},\alpha ), \end{aligned}$$

where

$$\begin{aligned}{} & {} \varUpsilon _{i,j}(\mathscr {A},\alpha )=\{z\in \mathbb {R}:(\mid z-\alpha _{i}\mid -R_{i}^{j}(\mathscr {A},\alpha _{i}))\mid z-\alpha _{j}\mid \\ {}{} & {} \le \mid a_{ij\ldots j}-\alpha _{i}e_{ij\ldots j}\mid R_{j}(\mathscr {A},\alpha _{j})\}, \end{aligned}$$

and

$$\begin{aligned} R_{i}^{j}(\mathscr {A},\alpha _{i})= R_{i}(\mathscr {A},\alpha _{i})-\mid a_{ij\ldots j}-\alpha _{i}e_{ij\ldots j}\mid . \end{aligned}$$

The remainder of this paper is organized as follows. In Sect. 2, we give a new Z-eigenvalues inclusion set with parameters and prove that it is tighter than that in Theorems 2 and 3. In Sect. 3, we consider two applications of the obtained Z-eigenvalue inclusion sets. The first application is to give a sufficient condition for positive definiteness of an even-order real symmetric tensors (also homogeneous polynomial forms). The second application is to judge the asymptotically stability of time-invariant polynomial systems. Finally, some concluding remarks are given to end this paper in Sect. 4.

2 Main Results

In this section, we give a new inclusion set $\varOmega (\mathscr {A},\alpha )$ and prove that it is tighter than the inclusion set $\mathscr {G}(\mathscr {A},\alpha )$ in Theorem 2 and the inclusion set $\varUpsilon (\mathscr {A},\alpha )$ in Theorem 3. Before giving the set $\varOmega (\mathscr {A},\alpha )$, we first give some notations and a lemma. Let

$$\begin{aligned} \varDelta =&\{(i_{2},\ldots ,i_{m}): i_{2}\ne \cdots \ne i_{m},\; \text{ or } \text{ only } \text{ two } \text{ of }\; i_{2},\ldots , i_{m}\in [n]\; \text{ are } \text{ the } \text{ same }\},\\ \overline{\varDelta }=&\{(i_{2},\ldots ,i_{m}): (i_{2},\ldots ,i_{m})\notin \varDelta , i_{2},\ldots , i_{m}\in [n]\},\\ N=&\{(i_{2},\ldots ,i_{m}): i_{2},\ldots , i_{m}\in [n]\}. \end{aligned}$$

Obviously,

$$\begin{aligned} \varDelta \cap \overline{\varDelta }=\emptyset ,\quad N=\varDelta \cup \overline{\varDelta },\; \text {and}\; \overline{\varDelta }=N\; \text{ when }\; \varDelta =\emptyset . \end{aligned}$$

Let

$$\begin{aligned} \varLambda _{i}=&\{(i_{2},\ldots ,i_{m}): e_{ii_{2}\ldots i_{m}}\ne 0,~ i_{2},\ldots , i_{m}\in [n]\},\\ \overline{\varLambda }_{i}=&\{(i_{2},\ldots ,i_{m}): e_{ii_{2}\ldots i_{m}}=0,~ i_{2},\ldots , i_{m}\in [n]\}, \end{aligned}$$

and

$$\begin{aligned} r_{i}^{\varDelta \cap \varLambda _{i}}(\mathscr {A},\alpha _i)= & {} \frac{1}{(m-2)^{\frac{m-2}{2}}}\sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{i}}\mid a_{ii_2\ldots i_m}-\alpha _i e_{ii_2\ldots i_m}\mid , \nonumber \\ r_{i}^{\overline{\varDelta }\cap \varLambda _{i}}(\mathscr {A},\alpha _i)= & {} \sum _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{i}}\mid a_{ii_2\ldots i_m}-\alpha _i e_{ii_2\ldots i_m}\mid , \nonumber \\ r_{i}^{\varDelta \cap \overline{\varLambda }_{i}}(\mathscr {A})= & {} \frac{1}{(m-2)^{\frac{m-2}{2}}}\sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{i} }\mid a_{ii_2\ldots i_m}\mid , \nonumber \\ r_{i}^{\overline{\varDelta }\cap \overline{\varLambda }_{i}}(\mathscr {A})= & {} \sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \overline{\varLambda }_{i} }\mid a_{ii_2\ldots i_m}\mid , \nonumber \\ r_{i}(\mathscr {A},\alpha _i)= & {} r_{i}^{\varDelta \cap \varLambda _{i}}(\mathscr {A},\alpha _i)+r_{i}^{\overline{\varDelta }\cap \varLambda _{i}}(\mathscr {A},\alpha _i) +r_{i}^{\varDelta \cap \overline{\varLambda }_{i}}(\mathscr {A})+r_{i}^{\overline{\varDelta }\cap \overline{\varLambda }_{i}}(\mathscr {A}). \end{aligned}$$

(2)

Then by $\frac{1}{(m-2)^{\frac{m-2}{2}}}\le 1$ for $m\ge 3$, it can be seen that

$$\begin{aligned} r_{i}(\mathscr {A},\alpha _i)\le R_{i}(\mathscr {A},\alpha _i),\quad i\in [n]. \end{aligned}$$

(3)

Lemma 1

[21, Lemma 2.2] Let $x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1$, where $x_{i}\in \mathbb {R},~i\in [n]$. If $y_{1}, y_{2},\ldots , y_{k}$ are arbitrary k entries of $x_{1}, x_{2}, \ldots ,x_{n}$, then

$$\begin{aligned} \mid y_{1}\mid \mid y_{2}\mid \cdots \mid y_{k}\mid \le \frac{1}{k^{\frac{k}{2}}}. \end{aligned}$$

Theorem 4

Let $\mathscr {A}=(a_{i_{1}\ldots i_{m}})\in {\mathbb {R}}^{[m,n]}$ with m even. Then for any $\alpha =(\alpha _1,\ldots ,\alpha _n)^{\top }\in \mathbb {R}^{n}$,

$$\begin{aligned} \sigma (\mathscr {A})\subseteq \varOmega (\mathscr {A},\alpha ) =\bigcup \limits _{i\in [n]}\bigcap \limits _{j\in [n],j\ne i}\varOmega _{i,j}(\mathscr {A},\alpha ), \end{aligned}$$

(4)

where

$$\begin{aligned} \varOmega _{i,j}(\mathscr {A},\alpha )=\{z\in \mathbb {R}: (|z-\alpha _i|-r_{i}(\mathscr {A},\alpha _i)+\mid a_{ij\ldots j}\mid )|z-\alpha _j|\le |a_{ij\ldots j}|r_j(\mathscr {A},\alpha _j) \}. \end{aligned}$$

Proof

Let $\lambda$ be any Z-eigenvalue of $\mathscr {A}$ and $x=(x_1,\ldots ,x_n)^{\top }\in \mathbb {R}^{n}\backslash \{0\}$ be a Z-eigenvector associated with $\lambda$. Then

$$\begin{aligned} \mathscr {A}x^{m-1}=\lambda x=\lambda \mathscr {E}x^{m-1}\quad and \quad ~x^\top x=1. \end{aligned}$$

(5)

Let $|x_t|$=$\max \nolimits _{i\in [n]}$ $|x_i|$. Then for any $s\in [n]$ and $s\ne t$, we have

$$\begin{aligned} (\lambda -\alpha _t) x_t =\,&\lambda x_t-\alpha _t x_t =\lambda x_t-\alpha _t \mathscr {E} x_t^{m-1}\\ =\,&\sum _{i_2,\ldots ,i_m\in [n]}a_{ti_2\ldots i_m}x_{i_2}\ldots x_{i_m}-\alpha _t \sum _{i_2,\ldots ,i_m\in [n]}e_{ti_2\ldots i_m}x_{i_2}\ldots x_{i_m}\\ =\,&\sum _{i_2,\ldots ,i_m\in [n]}(a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m})x_{i_2}\ldots x_{i_m}\\ =\,&\sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{t}}(a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m})x_{i_2}\ldots x_{i_m}\\&+\sum _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{t}}(a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m})x_{i_2}\ldots x_{i_m} \\ {}&+\sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{t}}a_{ti_2\ldots i_m}x_{i_2}\ldots x_{i_m}\\&+\sum \limits _{(i_2,\ldots ,i_m)\in (\overline{\varDelta }\cap \overline{\varLambda }_t)\backslash \{(s,\ldots ,s)\}}a_{ti_2\ldots i_m}x_{i_2}\ldots x_{i_m}+a_{ts\ldots s}x_s^{m-1}. \end{aligned}$$

Taking the modulus in above equation and using the triangle inequality and Lemma 1, we have

$$\begin{aligned}{} & {} |\lambda -\alpha _t||x_t| \le \, \sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{t}}\mid a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m}\mid \mid x_{i_2}\mid \ldots \mid x_{i_m}\mid \nonumber \\{} & {} +\sum _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{t}}\mid a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m}\mid \mid x_{i_2}\mid \ldots \mid x_{i_m}\mid \nonumber \\{} & {} +\sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{t}}\mid a_{ti_2\ldots i_m}\mid \mid x_{i_2}\mid \ldots \mid x_{i_m}\mid \nonumber \\{} & {} +\sum _{(i_{2},\ldots ,i_{m})\in (\overline{\varDelta }\cap \overline{\varLambda }_{t})\backslash \{(s,\ldots ,s)\}}\mid a_{ti_2\ldots i_m}\mid \mid x_{i_2}\mid \ldots \mid x_{i_m}\mid +\mid a_{ts\ldots s}\mid \mid x_s\mid ^{m-1}\nonumber \\ \le{} & {} \, \sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{t}}\mid a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m}\mid \mid y_{1}\mid \ldots \mid y_{m-2}\mid \mid x_{t}\mid \nonumber \\{} & {} +\sum _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{t}}\mid a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m}\mid \mid x_{t}\mid ^{m-1}\nonumber \\{} & {} +\sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{t}}\mid a_{ti_2\ldots i_m}\mid \mid z_{1}\mid \ldots \mid z_{m-2}\mid \mid x_{t}\mid \nonumber \\{} & {} +\sum _{(i_{2},\ldots ,i_{m})\in (\overline{\varDelta }\cap \overline{\varLambda }_{t})\backslash \{(s,\ldots ,s)\}}\mid a_{ti_2\ldots i_m}\mid \mid x_{t}\mid ^{m-1}+\mid a_{ts\ldots s}\mid \mid x_s\mid ^{m-1}\nonumber \\ \le{} & {} \, \frac{1}{(m-2)^{\frac{m-2}{2}}}\sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{t}}\mid a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m}\mid \mid x_{t}\mid \nonumber \\{} & {} +\sum _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{t}}\mid a_{ti_2\ldots i_m}-\alpha _t e_{ti_2\ldots i_m}\mid \mid x_{t}\mid \nonumber \\{} & {} +\frac{1}{(m-2)^{\frac{m-2}{2}}}\sum _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{t}}\mid a_{ti_2\ldots i_m}\mid \mid x_{t}\mid \nonumber \\{} & {} +\sum _{(i_{2},\ldots ,i_{m})\in (\overline{\varDelta }\cap \overline{\varLambda }_{t})\backslash \{(s,\ldots ,s)\}}\mid a_{ti_2\ldots i_m}\mid \mid x_{t}\mid +\mid a_{ts\ldots s}\mid \mid x_s\mid \nonumber \\ ={} & {} \, r_{t}^{\varDelta \cap \varLambda _{t}}(\mathscr {A},\alpha _t)\mid x_{t}\mid +r_{t}^{\overline{\varDelta }\cap \varLambda _{t}}(\mathscr {A},\alpha _t)\mid x_{t}\mid +r_{t}^{\varDelta \cap \overline{\varLambda }_{t}}(\mathscr {A})\mid x_{t}\mid \nonumber \\{} & {} +\left( r_{t}^{\overline{\varDelta }\cap \overline{\varLambda }_{t}}(\mathscr {A})-\mid a_{ts\ldots s}\mid \right) \mid x_{t}\mid +\mid a_{ts\ldots s}\mid \mid x_s\mid \nonumber \\ ={} & {} \,(r_{t}(\mathscr {A},\alpha _t)-\mid a_{ts\ldots s}\mid )\mid x_{t}\mid +\mid a_{ts\ldots s}\mid \mid x_s\mid , \end{aligned}$$

(6)

which implies that

$$\begin{aligned} (|\lambda -\alpha _t|-r_{t}(\mathscr {A},\alpha _t)+\mid a_{ts\ldots s}\mid )|x_t| \le |a_{ts\ldots s}| |x_s|. \end{aligned}$$

(7)

In (6), $|y_{1}|,\ldots ,|y_{m-2}|$ and $|z_{1}|,\ldots ,|z_{m-2}|$ are taken by the following two ways:

• (a) If $i_{2}\ne \ldots \ne i_{m}$, then we can enlarge any one of $|x_{2}|,\ldots ,|x_{m}|$ to $|x_t|$ and keep the others (can be taken as $|y_{1}|,\ldots ,|y_{m-2}|$ and $|z_{1}|,\ldots ,|z_{m-2}|$) unchanged.

• (b) If only two of $i_{2}, \ldots ,i_{m}$ are the same, then we can enlarge one of the two same elements to $|x_{t}|$ and keep others (can be taken as $|y_{1}|,\ldots ,|y_{m-2}|$ and $|z_{1}|,\ldots ,|z_{m-2}|$) unchanged.

If $|x_s|>0$ in (7), then from (5), we can get

$$\begin{aligned} (\lambda -\alpha _s) x_s =&\, \lambda x_s-\alpha _s \mathscr {E} x_s^{m-1}\\ =&\, {{\sum \limits _{{i_2,\ldots ,i_m \in [n]} }} a_{si_2\ldots i_m} x_{i_2}\ldots x_{i_m}}-\alpha _s{{\sum \limits _{{i_2,\ldots ,i_m \in [n]} }}}e_{si_2\ldots i_m}x_{i_2}\ldots x_{i_m}\\ =&\, \sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{s} }(a_{si_2\ldots i_m}-\alpha _s e_{si_2\ldots i_m}) x_{i_2}\ldots x_{i_m}\\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{s} }(a_{si_2\ldots i_m}-\alpha _s e_{si_2\ldots i_m}) x_{i_2}\ldots x_{i_m}\\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{s} }a_{si_2\ldots i_m} x_{i_2}\ldots x_{i_m} \\ {}&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \overline{\varLambda }_{s} }a_{si_2\ldots i_m} x_{i_2}\ldots x_{i_m}, \end{aligned}$$

and

$$\begin{aligned} |\lambda -\alpha _s||x_s| \le&\, \sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{s} }\mid a_{si_2\ldots i_m}-\alpha _s e_{si_2\ldots i_m}\mid \mid x_{i_2}\mid \ldots \mid x_{i_m}\mid \nonumber \\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{s} }\mid a_{si_2\ldots i_m}-\alpha _s e_{si_2\ldots i_m}\mid \mid x_{i_2}\mid \ldots \mid x_{i_m}\mid \nonumber \\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{s} }\mid a_{si_2\ldots i_m}\mid \mid x_{i_2}\mid \ldots \mid x_{i_m}\mid \nonumber \\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \overline{\varLambda }_{s} }\mid a_{si_2\ldots i_m}\mid \mid x_{i_2}\mid \ldots \mid x_{i_m}\mid \nonumber \\ \le&\, \sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{s} }\mid a_{si_2\ldots i_m}-\alpha _s e_{si_2\ldots i_m}\mid \mid y_{1}\mid \ldots \mid y_{m-2}\mid \mid x_{t}\mid \nonumber \\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{s} }\mid a_{si_2\ldots i_m}-\alpha _s e_{si_2\ldots i_m}\mid \mid x_{t}\mid ^{m-1}\nonumber \\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{s} }\mid a_{si_2\ldots i_m}\mid \mid z_{1}\mid \ldots \mid z_{m-2}\mid \mid x_{t}\mid \nonumber \\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \overline{\varLambda }_{s} }\mid a_{si_2\ldots i_m}\mid \mid x_{t}\mid ^{m-1}\nonumber \\ \le&\, \frac{1}{(m-2)^{\frac{m-2}{2}}}\sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \varLambda _{s} }\mid a_{si_2\ldots i_m}-\alpha _s e_{si_2\ldots i_m}\mid \mid x_{t}\mid \nonumber \\&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \varLambda _{s} }\mid a_{si_2\ldots i_m}-\alpha _s e_{si_2\ldots i_m}\mid \mid x_{t}\mid \nonumber \\&+\frac{1}{(m-2)^{\frac{m-2}{2}}}\sum \limits _{(i_{2},\ldots ,i_{m})\in \varDelta \cap \overline{\varLambda }_{s} }\mid a_{si_2\ldots i_m}\mid \mid x_{t}\mid \nonumber \\ {}&+\sum \limits _{(i_{2},\ldots ,i_{m})\in \overline{\varDelta }\cap \overline{\varLambda }_{s} }\mid a_{si_2\ldots i_m}\mid \mid x_{t}\mid \nonumber \\ =&\, r_{s}^{\varDelta \cap \varLambda _{t}}(\mathscr {A},\alpha _s)\mid x_{t}\mid +r_{s}^{\overline{\varDelta }\cap \varLambda _{t}}(\mathscr {A},\alpha _s)\mid x_{t}\mid \nonumber \\&+r_{s}^{\varDelta \cap \overline{\varLambda }_{s}}(\mathscr {A})\mid x_{t}\mid +r_{s}^{\overline{\varDelta }\cap \overline{\varLambda }_{s}}(\mathscr {A})\mid x_{t}\mid \nonumber \\ =&\, r_{s}(\mathscr {A},\alpha _s)\mid x_{t}\mid , \end{aligned}$$

(8)

which leads to

$$\begin{aligned} |\lambda -\alpha _s||x_s| \le r_s(\mathscr {A},\alpha _s)|x_t|. \end{aligned}$$

(9)

Note here that $|y_{1}|,\ldots ,|y_{m-2}|$ and $|z_{1}|,\ldots ,|z_{m-2}|$ in (8) are taken in the same way as in (6).

Multiplying (7) and (9) yields

$$\begin{aligned} (|\lambda -\alpha _t|-r_{t}(\mathscr {A},\alpha _t)+\mid a_{ts\ldots s}\mid )|\lambda -\alpha _s||x_t||x_s| \le |a_{ts\ldots s}|r_s(\mathscr {A},\alpha _s)|x_t||x_s|. \end{aligned}$$

Furthermore, by $\mid x_t\mid \mid x_s\mid >0$, we can get

$$\begin{aligned} (|\lambda -\alpha _t|-r_{t}(\mathscr {A},\alpha _t)+\mid a_{ts\ldots s}\mid )|\lambda -\alpha _s|\le |a_{ts\ldots s}|r_s(\mathscr {A},\alpha _s), \end{aligned}$$

(10)

which implies that

$$\begin{aligned} \lambda \in \varOmega _{t,s}(\mathscr {A},\alpha ). \end{aligned}$$

(11)

If $|x_s|=0$ in (7), by $|x_t|>0$, we have $|\lambda -\alpha _t|-r_t(\mathscr {A},\alpha _t)+\mid a_{ts\ldots s}\mid \le 0$, which implies that (10) holds, consequently, (11) holds.

By the arbitrariness of $s\in [n]$, $s\ne t$, we have

$$\begin{aligned} \lambda \in \bigcap \limits _{s\in [n], s\ne t}\varOmega _{t,s}(\mathscr {A},\alpha ). \end{aligned}$$

Furthermore, by the uncertainty of choosing $t\in [n]$, we have

$$\begin{aligned} \lambda \in \bigcup \limits _{t\in [n]}\bigcap \limits _{s\in [n], s\ne t}\varOmega _{t,s}(\mathscr {A},\alpha ). \end{aligned}$$

Consequently, $\sigma (\mathscr {A})\subseteq \varOmega (\mathscr {A},\alpha )$. $\square$

The following comparison theorem shows that the Z-eigenvalue inclusion set $\varOmega (\mathscr {A},\alpha )$ in Theorem 4 is tighter (that is, can capture all Z-eigenvalue of $\mathscr {A}$ more accurate ) than those in Theorems 2 and 3.

Theorem 5

Let $\mathscr {A}=(a_{i_{1}\ldots i_{m}})\in {\mathbb {R}}^{[m,n]}$. Then for any $\alpha =(\alpha _1,\ldots ,\alpha _n)^{\top }\in \mathbb {R}^{n}$,

$$\begin{aligned} \varOmega (\mathscr {A},\alpha )\subseteq \varUpsilon (\mathscr {A},\alpha )\subseteq \mathscr {G}(\mathscr {A},\alpha ). \end{aligned}$$

Proof

From Corollary 1 of [22], it can be seen that $\varUpsilon (\mathscr {A},\alpha )\subseteq \mathscr {G}(\mathscr {A},\alpha )$. Below we only need to prove $\varOmega (\mathscr {A},\alpha )\subseteq \varUpsilon (\mathscr {A},\alpha )$. Let $z \in \varOmega (\mathscr {A},\alpha )$. Then there are $i, j\in [n]$ and $i\ne j$ such that $z\in \varOmega _{i,j}(\mathscr {A},\alpha )$, i.e.,

$$\begin{aligned} (|\lambda -\alpha _i|-r_i(\mathscr {A},\alpha _i)+\mid a_{ij\ldots j}\mid )|\lambda -\alpha _j|\le |a_{ij\ldots j}|r_j(\mathscr {A},\alpha _j). \end{aligned}$$

(12)

For Cases I and II of the Z-identity tensor, we have $e_{ij\ldots j}=0$, that is,

$$\begin{aligned} \mid a_{ij\ldots j}-\alpha _te_{ij\ldots j}\mid =\mid a_{ij\ldots j}\mid . \end{aligned}$$

By (3), we have

$$\begin{aligned} R_i^j(\mathscr {A},\alpha _i)&= R_i(\mathscr {A},\alpha _i)-\mid a_{ij\ldots j}-\alpha _i e_{ij\ldots j}\mid \\ {}&= R_i(\mathscr {A},\alpha _i)-\mid a_{ij\ldots j}\mid \ge&r_i(\mathscr {A},\alpha _i)-\mid a_{ij\ldots j}\mid . \end{aligned}$$

By (12), we have

$$\begin{aligned} (\mid z-\alpha _i\mid -R_i^j(\mathscr {A},\alpha _i))\mid z-\alpha _j\mid \le&\, (|z-\alpha _i|-r_i(\mathscr {A},\alpha _i)+\mid a_{ij\ldots j}\mid )|z-\alpha _j|\\ \le&\, |a_{ij\ldots j}|r_j(\mathscr {A},\alpha _j)\\ \le&\, \mid a_{ij\ldots j}-\alpha _i e_{ij\ldots j}\mid R_j(\mathscr {A},\alpha _j), \end{aligned}$$

i.e.,

$$\begin{aligned} (\mid z-\alpha _{i}\mid -R_{i}^{j}(\mathscr {A},\alpha _{i}))\mid z-\alpha _{j}\mid \le \mid a_{ij\ldots j}-\alpha _{i}e_{ij\ldots j}\mid R_{j}(\mathscr {A},\alpha _{j}), \end{aligned}$$

which implies that $z\in \varUpsilon _{i,j}(\mathscr {A},\alpha )$. Hence, $\varOmega (\mathscr {A},\alpha )\subseteq \varUpsilon (\mathscr {A},\alpha )$. $\square$

As $r_{i}(\mathscr {A},\alpha _i)$, $i\in [n]$, are related to the Z-identity tensor $\mathscr {E}$ and the order and dimension of $\mathscr {A}$, we list the specific form of $\varOmega (\mathscr {A},\alpha )$ in Theorem 4 with $m=4$ and $m=6$ by using the similar methods as [21, Corollary 2 and Corollary 3] follows.

Corollary 1

Let $\mathscr {A}=(a_{ijkl})\in {\mathbb {R}}^{[4,n]}$. Then for any $\alpha =(\alpha _1,\ldots ,\alpha _n)^{\top }\in \mathbb {R}^{n}$, (4) holds, where $r_{i}(\mathscr {A},\alpha _i)$ are taken by the following two cases:

• (i) If the Z-identify tensor $\mathscr {E}$ is taken as $\mathscr {E}_1$, then

  $$\begin{aligned} r_{i}(\mathscr {A},\alpha _i){} & {} =\frac{1}{2}\sum \limits _{j\ne i}\mid a_{iijj}-\alpha _i\mid +\mid a_{iiii}-\alpha _i\mid \\ {}{} & {} \quad +\frac{1}{2}\Big (R_{i}(\mathscr {A})+\sum \limits _{j\ne i}\mid a_{ijjj}\mid -\sum \limits _{j\in [n]}\mid a_{iijj}\mid \Big ). \end{aligned}$$

• (ii) If the Z-identify tensor $\mathscr {E}$ is taken as $\mathscr {E}_2$, then

  $$\begin{aligned} r_{i}(\mathscr {A},\alpha _i){} & {} =\frac{1}{2}\sum \limits _{j\ne i}\Big (\mid a_{iijj}-\frac{1}{3}\alpha _i\mid +\mid a_{ijij}-\frac{1}{3}\alpha _i\mid +\mid a_{ijji}-\frac{1}{3}\alpha _i\mid \Big )\\{} & {} \quad +\mid a_{iiii}-\alpha _i\mid +\widetilde{r}_{i}(\mathscr {A}), \end{aligned}$$

  where

  $$\begin{aligned} \widetilde{r}_{i}(\mathscr {A})=\frac{1}{2}\Big (R_{i}(\mathscr {A})+\sum \limits _{j\ne i}\mid a_{ijjj}\mid -\sum \limits _{j\ne i}(\mid a_{iijj}\mid +\mid a_{ijij}\mid +\mid a_{ijji}\mid )-\mid a_{iiii}\mid \Big ). \end{aligned}$$

Corollary 2

Let $\mathscr {A}=(a_{i_1\ldots i_6})\in {\mathbb {R}}^{[6,n]}$. Then for any $\alpha =(\alpha _1,\ldots ,\alpha _n)^{\top }\in \mathbb {R}^{n}$, (4) holds, $r_{i}(\mathscr {A},\alpha _i)$ are taken by the following two cases:

• (i) If the Z-identify tensor $\mathscr {E}$ is taken as $\mathscr {E}_1$, then

  $$\begin{aligned} r_{i}(\mathscr {A},\alpha _i){} & {} =\sum \limits _{j,k\in [n]}\mid a_{iijjkk}-\alpha _i\mid \\{} & {} \quad +\left\{ \begin{array}{ll} R_{i}(\mathscr {A})-\sum \limits _{j,k\in [n]}\mid a_{iijjkk}\mid ,\quad 2\le n\le 3;\\ R_{i}(\mathscr {A})-\sum \limits _{j,k\in [n]}\mid a_{iijjkk}\mid -\frac{15}{16}\sum \limits _{(j,k,l,s,t)\in \varDelta }\mid a_{ijklst}\mid ,&{}\quad n\ge 4. \end{array} \right. \end{aligned}$$

• (ii) If the Z-identify tensor $\mathscr {E}$ is taken as $\mathscr {E}_2$, then (2) holds and

  $$\begin{aligned} r_{i}^{\varDelta \cap \varLambda _{i}}(\mathscr {A},\alpha _i)=&\, \frac{1}{16}\sum \limits _{j\ne k\ne i}\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,j,j,k,k))\}}\Big |a_{ii_2\ldots i_6}-\frac{1}{15}\alpha _{i}\Big |,\\ r_{i}^{\overline{\varDelta }\cap \varLambda _{i}}(\mathscr {A},\alpha _i)=&\, |a_{iiiiii}-\alpha _{i}| +\sum \limits _{j\ne i}\Bigg (\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,i,i,j,j))\}}\Big |a_{ii_2\ldots i_6}-\frac{1}{5}\alpha _{i}\Big |\\&+\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,j,j,j,j))\}}\Big |a_{ii_2\ldots i_6}-\frac{1}{5}\alpha _{i}\Big |\Bigg ),\\ r_{i}^{\varDelta \cap \overline{\varLambda }_{i}}(\mathscr {A})=&\, \frac{1}{16}\Bigg \{\sum \limits _{j\ne k\ne l\ne i} \Bigg (\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,j,j,k,l))\}}\Big |a_{ii_2\ldots i_6}\Big |\\&+\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,i,j,k,l))\}}\Big |a_{ii_2\ldots i_6}\Big |\\&+\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (j,j,k,k,l))\}}\Big |a_{ii_2\ldots i_6}\Big |\\ {}&+\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (j,j,j,k,l))\}}\Big |a_{ii_2\ldots i_6}\Big |\Bigg )\\&+\sum \limits _{j\ne k\ne l\ne s\ne i}\Bigg (\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,j,k,l,s))\}}\Big |a_{ii_2\ldots i_6}\Big |\\&+\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (j,j,k,l,s))\}}\Big |a_{ii_2\ldots i_6}\Big |\Bigg )\\&+\sum \limits _{j\ne k\ne l\ne s\ne p\ne i}\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (j,k,l,s,p))\}}\Big |a_{ii_2\ldots i_6}\Big |\Bigg \},\\ r_{i}^{\overline{\varDelta }\cap \overline{\varLambda }_{i}}(\mathscr {A})=&\,R_{i}(\mathscr {A})-|a_{iiiiii}|-16r_{i}^{\varDelta \cap \overline{\varLambda }_{i}}(\mathscr {A})\\ {}&-\sum \limits _{j\ne k\ne i} \sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,j,j,k,k))\}}\Big |a_{ii_2\ldots i_6}\Big |\\&-\sum \limits _{j\ne i}\Bigg (\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,i,i,j,j))\}}\Big |a_{ii_2\ldots i_6}\Big | \\ {}&+\sum \limits _{\{(i_2,\ldots ,i_6)\}\in \{(\pi (i,j,j,j,j))\}}\Big |a_{ii_2\ldots i_6}\Big |\Bigg ), \end{aligned}$$

  where $\{(\pi (i,j,k,l,s))\}$ represents the set of all permutations of indexes i, j, k, l, s.

3 Applications

In this section, two applications are considered. By using the inclusion set $\varOmega (\mathscr {A},\alpha )$ in Theorem 4, we give sufficient conditions for the positive definiteness of an even-order real symmetric tensor (also the homogeneous polynomial forms) and the asymptotically stability of time-invariant polynomial systems.

3.1 Positive Definiteness of Homogeneous Polynomial Forms

Based on the inclusion interval $\mathscr {G}(\mathscr {A},\alpha )$ in Theorem 2, Li et al. in [8] obtained a sufficient condition of the positive definiteness of an even-order tensor as follows.

Definition 1

[8, Definition 3.1] Let $\mathscr {A}\in \mathbb {R}^{[m,n]}$ with m even and $\alpha =(\alpha _1,\ldots ,\alpha _n)^{\top }\in \mathbb {R}^{n}$. We call $\mathscr {A}$ an $\alpha$-strictly diagonally dominant tensor of even order if

$$\begin{aligned} \alpha _i>R_{i}(\mathscr {A}, \alpha _{i}), \quad i\in [n]. \end{aligned}$$

(13)

Theorem 6

[8, Theorem 3.2] Let $\mathscr {A}=(a_{i_1\ldots i_m})\in \mathbb {R}^{[m,n]}$ with m even, and $\lambda$ be a Z-eigenvalue of $\mathscr {A}$. If $\mathscr {A}$ is an $\alpha$-strictly diagonally dominant tensor with all $\alpha _i>0$ for each $i\in [n]$, then $\lambda >0$. Furthermore, if $\mathscr {A}$ is also symmetric, then $\mathscr {A}$ is positive definite, consequently, f(x) defined in (1) is positive definite.

Based on the inclusion interval $\varUpsilon (\mathscr {A},\alpha )$ in Theorem 3, Shen et al. in [22] obtained a sufficient condition of the positive definiteness of an even-order weakly symmetric tensor as follows.

Theorem 7

[22, Theorem 3] Let $\lambda$ be a Z-eigenvalue of $\mathscr {A}=(a_{i_1\ldots i_m})\in \mathbb {R}^{[m,n]}$ and $\mathscr {E}\in \mathbb {R}^{[m,n]}$ be a Z-identity tensor. If there exists a positive real vector $\alpha =(\alpha _1,\ldots ,\alpha _n)^\top$ and $i,j\in [n]$ with $j\ne i$ such that

$$\begin{aligned} \left( \alpha _i-R_i^j(\mathscr {A},\alpha _i)\right) \alpha _j>\mid a_{ij\ldots j}-\alpha _i e_{ij\ldots j}\mid R_j(\mathscr {A},\alpha _j), \end{aligned}$$

(14)

then $\lambda >0$. Further, if $\mathscr {A}$ weakly symmetric, then $\mathscr {A}$ is positive definite and f(x) defined in (1) is positive definite.

Based on the inclusion interval $\varOmega (\mathscr {A},\alpha )$ in Theorem 4, a sufficient condition for the positive definiteness of an even-order tensor can be obtained.

Definition 2

Let $\mathscr {A}=(a_{i_{1}\ldots i_{m}})\in \mathbb {R}^{[m,n]}$ with m even. If there is $\alpha =(\alpha _1,\ldots ,\alpha _n)^{\top }\in \mathbb {R}^{n}$ such that for any $i,j\in [n]$ and $j\ne i$,

$$\begin{aligned} (|\alpha _i|-r_{i}(\mathscr {A},\alpha _i)+\mid a_{ij\ldots j}\mid )|\alpha _j| > |a_{ij\ldots j}|r_j(\mathscr {A},\alpha _j), \end{aligned}$$

(15)

then we call $\mathscr {A}$ an even-order double $\alpha$-strictly diagonally dominant tensor.

Now, the relationship between $\alpha$-strictly diagonally dominant tensors and double $\alpha$-strictly diagonally dominant tensors is discussed.

Theorem 8

Let $\mathscr {A}=(a_{i_1\ldots i_{m}})\in \mathbb {R}^{[m,n]}$. If $\mathscr {A}$ is an $\alpha$-strictly diagonally dominant tensor, then $\mathscr {A}$ is a double $\alpha$-strictly diagonally dominant tensor.

Proof

Let $\mathscr {A}$ be an $\alpha$-strictly diagonally dominant tensor. Then for each $i\in [n]$, we have

$$\begin{aligned} |\alpha _i| > R_{i}(\mathscr {A},\alpha _i)\ge r_{i}(\mathscr {A},\alpha _i), \end{aligned}$$

that is,

$$\begin{aligned} |\alpha _i| > r_{i}(\mathscr {A},\alpha _i), \end{aligned}$$

which implies that

$$\begin{aligned} |\alpha _i|-r_i(\mathscr {A}, \alpha _i)+|a_{ij\ldots j}|> |a_{ij\ldots j}|,\quad j\in [n],\quad j\ne i. \end{aligned}$$

(16)

For this index $j\in [n]$, we have

$$\begin{aligned} \mid \alpha _j\mid >r_{j}(\mathscr {A}, \alpha _j). \end{aligned}$$

(17)

Multiplying (16) and (17) yields (15), which implies that $\mathscr {A}$ is a double $\alpha$-strictly diagonally dominant tensor. $\square$

Theorem 9

Let $\mathscr {A}=(a_{i_{1}\ldots i_{m}})\in \mathbb {R}^{[m,n]}$ with m even and $\lambda$ be any Z-eigenvalue of $\mathscr {A}$. If there is positive vector $\alpha =(\alpha _1,\ldots ,\alpha _n)^{\top }\in \mathbb {R}^{n}$ such that $\mathscr {A}$ is a double $\alpha$-strictly diagonally dominant tensor, then $\lambda >0$. Furthermore, if $\mathscr {A}$ is symmetric, then $\mathscr {A}$ is positive definite and, consequently, f(x) is positive definite.

Proof

Suppose on the contrary that $\lambda \le 0$. According to Theorem 4, we have $\lambda \in \varOmega (\mathscr {A},\alpha )$, which implies that there are $i,j\in [n]$ and $i\ne j$ such that $\lambda \in \varOmega _{i,j}(\mathscr {A},\alpha )$, i.e.,

$$\begin{aligned} (|\lambda -\alpha _i|-r_{i}(\mathscr {A},\alpha _i)+\mid a_{ij\ldots j}\mid )|\lambda -\alpha _j|\le |a_{ij\ldots j}|r_j(\mathscr {A},\alpha _j). \end{aligned}$$

(18)

On the other hand, by $\alpha _i>0$, $\alpha _j>0$, $\lambda \le 0$ and (15), it follows that

$$\begin{aligned} (|\lambda -\alpha _i|-r_i(\mathscr {A},\alpha _i)+|a_{ij\ldots j}|)|\lambda -\alpha _j| \ge&\, (|\alpha _i|-r_{i}(\mathscr {A},\alpha _i)+\mid a_{ij\ldots j}\mid )|\alpha _j| \nonumber \\ >&\, |a_{ij\ldots j}|r_j(\mathscr {A},\alpha _j). \end{aligned}$$

(19)

It is easy to see that (18) and (19) contradict each other. Consequently, $\lambda >0$. Furthermore, if $\mathscr {A}$ is symmetric tensor, then all Z-eigenvalues of $\mathscr {A}$ are positive, which implies that $\mathscr {A}$ is positive definite and, consequently, f(x) is positive definite. $\square$

Finally, an example is given to verify the effectiveness of Theorem 9. Before that, a lemma is recalled.

Lemma 2

[21, Lemma 4.2] Let

$$\begin{aligned} g(x)=x-\frac{1}{a}\sum \limits _{i\in [n]}|x-b_i|-c \end{aligned}$$

be a real-valued function about x, where a is a positive integer, $b_i\in \mathbb {R}$ and $b_1\le b_2 \le \ldots \le b_{n}$ with $n \ge a$, and $c\in \mathbb {R}$.

• (I) Assume that a is odd.

• (I.i) If n is odd, then

  $$\begin{aligned} \max \limits _{x\in \mathbb {R}}g(x)=\frac{1}{a}\left( \sum \limits _{i=1}^{\frac{n+a}{2}}b_i-\sum \limits _{i=\frac{n+a}{2}+1}^n b_i\right) -c, \end{aligned}$$

  (20)

  and this takes place for every $x\in [b_{\frac{n+a}{2}},b_{\frac{n+a}{2}+1}]$ if $b_{\frac{n+a}{2}} \ne b_{\frac{n+a}{2}+1}$, and only for $x=b_{\frac{n+a}{2}}$ if $b_{\frac{n+a}{2}}=b_{\frac{n+a}{2}+1}$. Note that let $[b_{\frac{n+a}{2}},b_{\frac{n+a}{2}+1}]$ be $[b_{\frac{n+a}{2}},+\infty )$ if $b_{\frac{n+a}{2}+1}$ does not exist.

• (I.ii) If n is even, then

  $$\begin{aligned} \max \limits _{x\in \mathbb {R}} g(x)=\frac{1}{a}\left( \sum \limits _{i=1}^{\frac{n+a-1}{2}}b_i-\sum \limits _{i=\frac{n+a+3}{2}}^n b_i\right) -c, \end{aligned}$$

  (21)

  and this maximum is reached when $x=b_{\frac{n+a+1}{2}}$.

• (II) Assume that a is even. If n is odd, then (21) holds. And if n is even, then (20) holds.

Example 1

Let $\mathscr {A}=(a_{ijkl})\in {\mathbb {R}}^{[4,3]}$ with entries defined as follows:

$$\begin{aligned} a_{1111}= & {} 2.6,\quad a_{2222}=3.2,\quad a_{3333}=2,\quad a_{1112}=a_{1121}=a_{1211}=a_{2111}=0.4,\\ a_{1122}= & {} a_{1212}=a_{1221}=a_{2112}=a_{2121}=a_{2211}=0.9,\\ a_{1133}= & {} a_{1313}=a_{1331}=a_{3113}=a_{3131}=a_{3311}=1.1,\\ a_{1233}= & {} a_{1323}=a_{1332}=a_{2133}=a_{2313}=a_{2331}=0.4,\\ a_{3123}= & {} a_{3132}=a_{3213}=a_{3231}=a_{3312}=a_{3321}=0.3,\\ a_{2223}= & {} a_{2232}=a_{2322}=a_{3222}=0.4,\\ a_{2233}= & {} a_{2323}=a_{2332}=a_{3223}=a_{3232}=a_{3322}=1, \end{aligned}$$

and $a_{ijkl}=0$ for otherwise.

Our goal is to judge the positive definiteness of $\mathscr {A}$. Because of the form of $R_i(\mathscr {A},\alpha _i)$ and $r_i(\mathscr {A},\alpha _i)$ being related to the Z-identify tensor $\mathscr {E}$, we now divide two cases to consider the positive definiteness of $\mathscr {A}$.

Case I. Let the Z-identify tensor $\mathscr {E}$ be $\mathscr {E}_1=(e_{ijkl})\in \mathbb {R}^{[4,3]}$, i.e.,

$$\begin{aligned} e_{1111}=e_{1122}=e_{1133}=e_{2222}=e_{2211}=e_{2233}=e_{3333}=e_{3311}=e_{3322}=1, \end{aligned}$$

and $e_{ijkl}=0$ for otherwise.

Proposition 1 of [21] shows that Theorem 6 cannot be used to judge the positive definiteness of $\mathscr {A}$ when the Z-identify tensor $\mathscr {E}$ in $R_i(\mathscr {A},\alpha _i)$ is taken as $\mathscr {E}_1$. Now, we consider using Theorem 7 to judge the positive definiteness of $\mathscr {A}$. Let $\alpha =(2.5, 4, 2)^\top$. By

$$\begin{aligned} (\alpha _2-R_2^1(\mathscr {A},\alpha _2))\alpha _1=-4.25<1.88=\mid a_{2111}-\alpha _2 e_{2111}\mid R_1(\mathscr {A},\alpha _1), \end{aligned}$$

it can be seen that (14) does not hold for $i=2$ and $j=1$, which implies that we cannot use Theorem 7 to judge the positive definiteness of $\mathscr {A}$ for this $\alpha$. However, for this $\alpha$, we can use Theorem 9 to judge the positive definiteness of $\mathscr {A}$. In fact, by the numerical results of (15) listed in Table 1, it can be seen that (15) holds for all $i,j\in [n]$ and $j\ne i$, which implies that we can use Theorem 9 to judge the positive definiteness of $\mathscr {A}$.

Table 1 Numerical results of (15) with the Z-identify tensor $\mathscr {E}_1$

We also use the Z-eigenvalue inclusion sets to judge the positive definiteness of $\mathscr {A}$. By Theorem 1, we have

$$\begin{aligned} \mathscr {L}(\mathscr {A})=\left\{ z\in \mathbb {C}:\mid z\mid \le \frac{5.9+\sqrt{45.21}}{2}\right\} . \end{aligned}$$

By Theorem 2, we have

$$\begin{aligned} \mathscr {G}(\mathscr {A},\alpha )=\{z\in \mathbb {R}:\mid z-4\mid \le 13.5\}. \end{aligned}$$

By Theorem 3, we have

$$\begin{aligned} \varUpsilon (\mathscr {A},\alpha )=\{z\in \mathbb {R}:(\mid z-4\mid -5.7)\mid z-2.5\mid \le 1.88\}. \end{aligned}$$

By Theorem 4, we have

$$\begin{aligned} \varOmega (\mathscr {A},\alpha )=\{z\in \mathbb {R}:(\mid z-4\mid -3.25)\mid z-2.5\mid \le 0.96\}. \end{aligned}$$

The Z-eigenvalue inclusion sets $\mathscr {L}(\mathscr {A})$, $\mathscr {G}(\mathscr {A},\alpha )$, $\varUpsilon (\mathscr {A},\alpha )$, $\varOmega (\mathscr {A},\alpha )$ and the exact Z-eigenvalues are drawn in Fig. 1, where they are represented by black dotted boundary, green stippled boundary, blue dotted boundary, red solid boundary and black “$+$,” respectively.

Fig. 1

Comparisons of $\mathscr {L}(\mathscr {A})$, $\mathscr {G}(\mathscr {A},\alpha )$, $\varUpsilon (\mathscr {A},\alpha )$ and $\varOmega (\mathscr {A},\alpha )$ with $\mathscr {E}_1$ and $\alpha =(2.5, 4, 2)^\top$

Case II. Let the Z-identify tensor $\mathscr {E}$ be $\mathscr {E}_2=(e_{ijkl})\in \mathbb {R}^{[4,3]}$, i.e.,

$$\begin{aligned} e_{1111}=e_{2222}=e_{3333}=1,~~ e_{1122}=e_{1212}=e_{1221}=\frac{1}{3},~~ e_{1133}=e_{1313}=e_{1331}=\frac{1}{3}, \end{aligned}$$

and $e_{ijkl}=0$ for otherwise.

Then

$$\begin{aligned}{} & {} R_{i}(\mathscr {A}, \alpha _{i})\\ {}{} & {} \qquad =\mid a_{iiii}-\alpha _{i}\mid +\sum \limits _{j\ne i}\left( \mid a_{iijj}-\frac{1}{3}\alpha _{i}\mid +\mid a_{ijij}-\frac{1}{3}\alpha _{i}\mid +\mid a_{ijji}-\frac{1}{3}\alpha _{i}\mid \right) +\gamma _{i}, \end{aligned}$$

where

$$\begin{aligned} \gamma _{i}=R_{i}(\mathscr {A})-\mid a_{iiii}\mid -\sum \limits _{j\ne i}(\mid a_{iijj}\mid +\mid a_{ijij}\mid +\mid a_{ijji}\mid ),\quad i\in [3]. \end{aligned}$$

Firstly, we use Theorem 6 to judge the positive definiteness of $\mathscr {A}$. Suppose that there is $\alpha =(\alpha _1,\alpha _2,\alpha _3)^{\top }\in \mathbb {R}^{3}$ such that (13) holds, which implies that

$$\begin{aligned} f(\alpha _i): =&\, \alpha _i-|a_{iiii}-\alpha _i|-\sum \limits _{j\ne i}\left( |a_{iijj}-\frac{1}{3}\alpha _{i}|+|a_{ijij}-\frac{1}{3}\alpha _i|+|a_{ijji}-\frac{1}{3}\alpha _i|\right) \\ =&\, \alpha _i-\frac{1}{3}\left[ 3|\alpha _i-a_{iiii}|+\sum \limits _{j\ne i}(|\alpha _i-3a_{iijj}|+|\alpha _i-3a_{ijij}|+|\alpha _i-3a_{ijji}|)\right] \\ >&\gamma _i. \end{aligned}$$

By Lemma 2, we have

$$\begin{aligned} \max \limits _{\alpha _{1}\in \mathbb {R}}f(\alpha _1)=2< & {} 2.4=\gamma _{1},\\ \max \limits _{\alpha _{2}\in \mathbb {R}}f(\alpha _2)=2.5< & {} 2.8=\gamma _{2},\\ \max \limits _{\alpha _{3}\in \mathbb {R}}f(\alpha _3)=1.7< & {} 2.2=\gamma _{3}, \end{aligned}$$

which shows that there is not $\alpha _1$, $\alpha _2$ and $\alpha _3$ such that (13) holds and implies that we cannot use Theorem 6 to judge the positive definiteness of $\mathscr {A}$.

Secondly, we use Theorem 7 to judge the positive definiteness of $\mathscr {A}$. Let $\alpha =(2.5, 3, 6)^\top$. By

$$\begin{aligned} (\alpha _3-R_3^2(\mathscr {A},\alpha _3))\alpha _2=-3 < 0.56=\mid a_{3222}-\alpha _3 e_{3222}\mid R_2(\mathscr {A},\alpha _2), \end{aligned}$$

it can be seen that (14) does not hold for $i=3$ and $j=1$, which implies that we cannot use Theorem 7 to judge the positive definiteness of $\mathscr {A}$.

However, for this $\alpha =(2.5, 3, 6)^\top$, we can use Theorem 9 to judge the positive definiteness of $\mathscr {A}$. The numerical results of (15) are listed in Table 2. From Table 2, it can be seen that (15) holds for all $i,j\in [n]$ and $j\ne i$, which implies that we can use Theorem 9 to judge the positive definiteness of $\mathscr {A}$. In fact, all Z-eigenvalues of $\mathscr {A}$ are 2.0000, 2.0035, 2.0224, 2.1335, 2.2539, 3.2022, 3.4147 and 3.7271.

Table 2 Numerical results of (15) with the Z-identify tensor $\mathscr {E}_2$

We also use the Z-eigenvalue inclusion sets to judge the positive definiteness of $\mathscr {A}$. Let $\alpha =(2.5, 3, 6)^\top$. By Theorem 2, we have

$$\begin{aligned} \mathscr {G}(\mathscr {A},\alpha )=\{z\in \mathbb {R}:\mid z-6\mid \le 11.9\}. \end{aligned}$$

By Theorem 3, we have

$$\begin{aligned} \varUpsilon (\mathscr {A},\alpha )=\{z\in \mathbb {R}:(\mid z-6\mid -6.7)\mid z-2.5\mid \le 0\}. \end{aligned}$$

By Theorem 4, we have

$$\begin{aligned} \varOmega (\mathscr {A},\alpha )=\{z\in \mathbb {R}:(\mid z-6\mid -5.35)\mid z-2.5\mid \le 0\}. \end{aligned}$$

The Z-eigenvalue inclusion sets $\mathscr {L}(\mathscr {A})$, $\mathscr {G}(\mathscr {A},\alpha )$, $\varUpsilon (\mathscr {A},\alpha )$, $\varOmega (\mathscr {A},\alpha )$ and the exact Z-eigenvalues are drawn in Fig. 2, where they are represented by black dotted boundary, green stippled boundary, blue dotted boundary, red solid boundary and black “$+$,” respectively.

Fig. 2

Comparisons of $\mathscr {L}(\mathscr {A})$, $\mathscr {G}(\mathscr {A},\alpha )$, $\varUpsilon (\mathscr {A},\alpha )$ and $\varOmega (\mathscr {A},\alpha )$ with $\mathscr {E}_2$ and $\alpha =(2.5, 3, 6)^\top$

From Figs. 1 and 2, it is easy to see that:

• (i) $0\in \mathscr {L}(\mathscr {A})$, $0\in \mathscr {G}(\mathscr {A},\alpha )$ and $0\in \varUpsilon (\mathscr {A},\alpha )$; consequently, the sets $\mathscr {L}(\mathscr {A})$, $\mathscr {G}(\mathscr {A},\alpha )$ and $\varUpsilon (\mathscr {A},\alpha )$ cannot be used to judge the positive definiteness of $\mathscr {A}$.

• (ii) $\sigma (\mathscr {A})\subseteq \varOmega (\mathscr {A},\alpha )\subset \mathbb {C}^+$, where $\mathbb {C}^+$ denotes the set of all complex numbers with positive real part, which implies that all Z-eigenvalues of $\mathscr {A}$ are positive and hence $\mathscr {A}$ is positive definite.

• (iii) $\sigma (\mathscr {A})\subseteq \varOmega (\mathscr {A},\alpha )\subseteq \varUpsilon (\mathscr {A},\alpha )\subseteq \mathscr {G}(\mathscr {A},\alpha )$, and $\mathscr {L}(\mathscr {A})$ and $\varOmega (\mathscr {A},\alpha )$ do not contain each other.

This example shows that no matter we take the Z-identify tensor $\mathscr {E}$ as $\mathscr {E}_1$ or $\mathscr {E}_2$, we can use Theorems 4 and 9 to judge the positive definiteness of the even-order tensors.

3.2 Asymptotically Stability of Time-Invariant Polynomial Systems

Consider the asymptotically stability of the time-invariant polynomial system

$$\begin{aligned} \varSigma :\dot{x}=\mathscr {A}^{(2)}x+\mathscr {A}^{(4)}x^3+\cdots +\mathscr {A}^{(2k)}x^{2k-1}, \end{aligned}$$

(22)

where $\mathscr {A}^{(t)}=(a_{i_1\ldots i_t})\in \mathbb {R}^{[t,n]}$, $t=2,4,\ldots ,2k$, and $x=(x_1,\ldots ,x_n)^\top$; see [1, 2]. A sufficient condition such that the nonlinear system (22) above is asymptotically stable is gave by Deng et al. in [1] as follows.

Theorem 10

[1, Theorem 3.3] For the nonlinear system $\varSigma$ in (22), if $-\mathscr {A}^{(t)}$ is positive definite, where $t=2,4,\ldots ,2k$, then the equilibrium point of $\varSigma$ is asymptotically stable.

By Theorems 9 and 10, a sufficient condition for the asymptotically stability can be given.

Theorem 11

For the nonlinear system $\varSigma$ in (22), if $-\mathscr {A}^{(t)}$ satisfies all conditions of Theorem 9, where $t=2,4,\ldots ,2k$, then the equilibrium point of $\varSigma$ is asymptotically stable.

Example 2

Consider the following polynomial system

$$\begin{aligned} \varSigma :\dot{x}_1= & {} -5x_1+2x_2+2x_3-2.6x_{1}^{3}-0.9x_{1}^{2}x_{2}-3.3x_{1}x_{3}^{2}-2.7x_{1}x_{2}^{2}-1.8x_{2}x_{3}^{2},\\ \dot{x}_2= & {} 2x_1-5x_2+2x_3-0.5x_{1}^{3}-3.2x_{2}^{3}-0.6x_{2}^{2}x_{3}\\ {}{} & {} -3.0x_{2}x_{3}^{2}-2.7x_{1}^{2}x_{2}-1.8x_{1}x_{3}^{2},\\ \dot{x}_3= & {} 2x_1+2x_2-5x_3-0.2x_{2}^{3}-2.4x_{3}^{3}-3.3x_{1}^{2}x_{3}-3.0x_{2}^{2}x_{3}-3.6x_{1}x_{2}x_{3}. \end{aligned}$$

Apparently, $\varSigma$ can be written as $\dot{x}=\mathscr {A}^{(2)}x+\mathscr {A}^{(4)}x^{3}$, where $x=(x_1,x_2,x_3)^{\top }$,

$$\begin{aligned} \mathscr {A}^{(2)}=\left( \begin{array}{ccc} -5&{}\quad 2&{}\quad 2\\ 2&{}\quad -5&{}\quad 2\\ 2&{}\quad 2&{}\quad -5 \end{array}\right) \end{aligned}$$

and $\mathscr {A}^{(4)}=(a_{ijkl})\in \mathbb {R}^{[4,3]}$ whose entries are as follows:

$$\begin{aligned} a_{1111}= & {} -2.6, \quad a_{2222}=-3.2,\quad a_{3333}=-2.4,\\ a_{1112}= & {} a_{1121}=a_{1211}=a_{2111}=-0.3,\\ a_{1122}= & {} a_{1212}=a_{1221}=a_{2112}=a_{2121}=a_{2211}=-0.9,\\ a_{1133}= & {} a_{1313}=a_{1331}=a_{3113}=a_{3131}=a_{3311}=-1.1,\\ a_{1233}= & {} a_{1323}=a_{1332}=a_{2133}=a_{2313}=a_{2331}=-0.6,\\ a_{3123}= & {} a_{3132}=a_{3213}=a_{3231}=a_{3312}=a_{3321}=-0.6,\\ a_{2223}= & {} a_{2232}=a_{2322}=a_{3222}=-0.2,\\ a_{2233}= & {} a_{2323}=a_{2332}=a_{3223}=a_{3232}=a_{3322}=-1.0, \end{aligned}$$

and $a_{ijkl}=0$ for otherwise.

It is easy to see that $-\mathscr {A}^{(2)}$ is positive definite. Now, we judge the positive definiteness of $-\mathscr {A}^{(4)}$ by taking the Z-identify tensor $\mathscr {E}$ as $\mathscr {E}_1$ and $\mathscr {E}_2$.

Case I. Let the Z-identify tensor $\mathscr {E}$ be $\mathscr {E}_1=(e_{ijkl})\in \mathbb {R}^{[4,3]}$.

Taking $\alpha =(3, 4, 2.8)^{\top }$, the numerical results of (15) are listed in Table 3. From Table 3, it can be seen that (15) holds for all $i,j\in [3]$ and $i\ne j$. Hence, $-\mathscr {A}^{(4)}$ is positive definite by Theorem 9.

Table 3 Numerical results of (15) with the Z-identify tensor $\mathscr {E}_1$ for $-\mathscr {A}^{(4)}$

Case II. Let the Z-identify tensor $\mathscr {E}$ be $\mathscr {E}_2=(e_{ijkl})\in \mathbb {R}^{[4,3]}$.

Taking $\alpha =(5, 3, 2)^{\top }$, the numerical results of (15) are listed in Table 4. From Table 4, it can be seen that (15) holds for all $i,j\in [3]$ and $i\ne j$, which implies that $-\mathscr {A}^{(4)}$ is positive definite by Theorem 9.

Table 4 Numerical results of (15) with the Z-identify tensor $\mathscr {E}_2$ for $-\mathscr {A}^{(4)}$

All in all, no matter the Z-identify tensor $\mathscr {E}$ is taken as $\mathscr {E}_1$ or $\mathscr {E}_2$, we can both judge the positive definiteness of $-\mathscr {A}^{(4)}$ by Theorem 9. In fact, all Z-eigenvalues of $-\mathscr {A}^{(4)}$ are 1.8287, 1.9538, 2.2721, 2.4000, 3.0829, 3.0838, 3.0954, 3.8162 and 3.9610. Furthermore, by Theorem 11, the equilibrium point of $\varSigma$ is asymptotically stable.

4 Conclusions

In this paper, we firstly presented a new Z-eigenvalue inclusion set $\varOmega (\mathscr {A},\alpha )$ in Theorem 4. Subsequently, we in Theorem 5 proved that it is tighter than the inclusion set $\mathscr {G}(\mathscr {A},\alpha )$ in Theorem 2.2 of [8] and the inclusion set $\varUpsilon (\mathscr {A},\alpha )$ in Theorem 1 of [22]. As an application of the new set $\varOmega (\mathscr {A},\alpha )$, we obtained a sufficient condition for the positive definiteness of an even-order real symmetric tensor (also homogeneous polynomial forms) in Theorem 9 and obtained a sufficient condition for the asymptotically stability of time-invariant polynomial systems in Theorem 11. Finally, we used Examples 1 and 2 to verify the validity of Theorems 9 and 11.

However, how to choose appropriate parameter vector $\alpha$ to minimize the Z-eigenvalue inclusion set $\varOmega (\mathscr {A},\alpha )$ in Theorem 4 is still an unsolved problem. We will continue to study this problem in the future.