Some new criteria for judging $\mathcal{H}$-tensors and their applications

1.   Introduction

Let $n$ and $m$ be integer numbers, $N = \{1, 2, \dots, n\}$ and $\mathbb{C} (\mathbb{R})$ be the set of all complex (real) numbers. A tensor $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})$ is called a complex (real) order $m$ dimension $n$ tensor, if $a_{i_{1}i_{2}\cdots i_{m}}\in \mathbb{C} (\mathbb{R})$, where $i_j = 1, 2, \dots, n$ for $j = 1, 2, \dots, m$. Let ${\mathbb{C}}^{\lbrack m, n\rbrack}$ $({\mathbb{R}}^{\lbrack m, n\rbrack})$ be the set of all complex (real) order $m$ dimension $n$ tensors. A tensor ${\mathcal{I}} = (\delta_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$ is called the unit tensor ^[1], if its elements satisfy

A tensor ${\mathcal{A}} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$ is called symmetric if

where $\Pi_m$ is the permutation group of $m$ indices.

At present, positive definite homogeneous polynomials play a critical role in the field of dynamics, and its positive definiteness can be transformed to identify the positive definiteness of the symmetric tensor associated with it ^[2]. However, for a given symmetric tensor, it is difficult to determine whether it is positive definite or not because the problem is NP-hard ^[3]. Thus, finding effective criteria to identify the positive definitiveness of a tensor is interesting.

$\mathcal{H}$-tensor was showed, Li et al. ^[3], that is a special kind of tensors in 2014 and an even-order symmetric $\mathcal{H}$-tensors with positive diagonal entries is positive definite. After that, some methods that judge the positive definiteness of a given tensor have been established ^{[4,5,6,7,8,9,10,11,12,13,14,15,16]}. Nevertheless, as presented by their range of judgment was fixed for the given tensor whether it was positive definite or not ^[14,15,16].

In this paper, some new criteria which only depend on elements of the given tensors are proposed to judge $\mathcal{H}$-tensors; they expand the range of judgment by an increasing constant $k$ which scales the elements of a given tensor. In addition, these criteria are used to judge the positive definiteness for even-order real symmetric tensors.

For the convenience of discussion, we start with the following notations, definitions and lemmas. The calligraphy letters $\mathcal{A}$, $\mathcal{B}$, $\cdots$ represent the tensors; the capital letters $A$, $B$, $\cdots$ denote the matrices; the lowercase letters $x$, $y$, $\cdots$ refer to the vectors.

For a tensor $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack} (m, n\geq 2)$, we denote

$r_{i}(\mathcal{A}) = \sum\limits_{\substack{i_{2}\cdots i_{m} \in N^{m-1} \\ \delta_{ii_{2}\cdots i_{m}} = 0}}|a_{ii_{2}\cdots i_{m}}| = \sum\limits_{\substack{i_{2}\cdots i_{m} \in N^{m-1}}}|a_{ii_{2}\cdots i_{m}}|-|a_{ii\cdots i}|,$

$N_1 = \{i\in N:|a_{ii\cdots i}| > r_{i}(\mathcal{A})\}$, $N_2 = \{i\in N:|a_{ii\cdots i}|\leq r_{i}(\mathcal{A})\}$,

$N_1^{m-1} = \{i_{2}i_{3}\cdots i_{m}:i_{j}\in N_1, j = 2, 3, \ldots, m\}$,

$N^{m-1}\backslash N_1^{m-1} = \{i_{2}i_{3}\cdots i_{m}:i_{2}i_{3}\cdots i_{m}\in N^{m-1}\; and\; i_{2}i_{3}\cdots i_{m}\notin N_1^{m-1}\}$,

$r_0 = 1$, $r_1 = \max\limits_{i\in N_1}\left\{\frac{r_{i}(\mathcal{A})}{|a_{ii\cdots i}|}\right\}$, $\cdots$,

$r_{k+1} = \max\limits_{i\in N_1}\left\{\frac{\sum\limits_{\substack{i_{2}\cdots i_{m} \in N^{m-1}\backslash N_1^{m-1}}} |a_{ii_{2}\cdots i_{m}}| +r_{k}\sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1} \\ \delta_{ii_{2}\cdots i_{m}} = 0}} |a_{ii_{2}\cdots i_{m}}|}{|a_{ii\cdots i}|} \right\}$, $k = 0, 1, 2, \dots$,

$\sigma _{k+1, i} = \frac{\sum\limits_{\substack{i_{2}\cdots i_{m} \in N^{m-1}\backslash N_1^{m-1}}} |a_{ii_{2}\cdots i_{m}}| +r_{k}\sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1} \\ \delta_{ii_{2}\cdots i_{m}} = 0}} |a_{ii_{2}\cdots i_{m}}|}{|a_{ii\cdots i}|}, \; i\in N_1, \; k = 0, 1, 2, \dots$.

It is obvious that we obtain $\sigma_{k+1, i}\leq r_{k+1}\leq r_k\leq \cdots\leq r_1 < r_0, \; i\in N_1, \; k = 0, 1, 2, \dots$.

Definition 1. ^[17] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack} (m, n\geq 2)$. If there is a positive vector $x = (x_1, x_2, \cdots, x_n)^T\in {\mathbb{R}^n}$ such that

where $|a|$ for the modulus of $a \in \mathbb{C}$ ^[17], then $\mathcal{A}$ is called an $\mathcal{H}$-tensor.

Definition 2. ^[18] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack} (m, n\geq 2)$. If

then $\mathcal{A}$ is called a strictly diagonally dominant tensor.

Definition 3. ^[8] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack} (m, n\geq 2)$ and $X = diag(x_1, x_2, \cdots, x_n)$. If

where

then we call $\mathcal{B}$ as the product of the tensor $\mathcal{A}$ and the matrix $X$.

Definition 4. ^[5] The product of $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack} (m, n\geq 2)$ and an $n$-by-$n$ matrrix $X = (x_{ij})$ on mode-$k$ is defined by

Definition 5. ^[5] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$. If there exists a $\varnothing \neq S \subset N$ such that $a_{i_{1}i_{2}\cdots i_{m}} = 0$, $\forall i_1\in S$ and $i_2, \ldots, i_m\notin S$, then $\mathcal{A}$ is called reducible. Otherwise, $\mathcal{A}$ is called irreducible.

Definition 6. ^[19] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$, for $i, j\in N$ and $i\neq j$, if there exists indices $k_1, k_2, \ldots, k_l$ with

where $k_0 = i$, $k_{l+1} = j$, we say that there is a nonzero element chain from $i$ to $j$.

Definition 7. ^[8] Let ${\mathcal{A}} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$; if the homogeneous polynomical equations satisfy:

then $\lambda$ is called an eigenvalue of $\mathcal{A}$ and $x$ is its corresponding eigenvector, where $\mathcal{A}x^{m-1}$ and $\lambda x^{\lbrack m-1 \rbrack}$ are vectors, and whose $i$ th components are

and

Definition 8. ^[20] For an $m$th degree homogeneous polynomial of $n$ variables, $f(x)$ can usually be denoted as

where $x = (x_1, x_2, \cdots, x_n)^T \in {\mathbb{R}^n}$. The homogeneous polynomial $f(x)$ can be represented as the tensor product of a symmetric tensor $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}$ and $x^m$ denoted by

where $x = (x_1, x_2, \cdots, x_n)^T \in {\mathbb{R}^n}$ ^[18]. If $m$ is even and

then we say that $f(x)$ is positive definite.

Lemma 1. ^[17] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$. $\mathcal{A}$ is an $\mathcal{H}$-tensor if $\mathcal{A}$ is a strictly diagonally dominant tensor.

Lemma 2. ^[3] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$. $\mathcal{A}$ is an $\mathcal{H}$-tensor if

● ($i$) $\mathcal{A}$ is irreducible;

● ($ii$) $|a_{ii\cdots i}|\geq r_{i}(\mathcal{A})$ for each $i\in N$;

● ($iii$) For the inequality of (ii), strict inequality holds for at least one $i$.

Lemma 3. ^[8] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$. $\mathcal{A}$ is an $\mathcal{H}$-tensor, if

● ($i$) $|a_{ii\cdots i}|\geq r_{i}(\mathcal{A}), \; i\in N$;

● ($ii$) $N_1 = \{i\in N:|a_{ii\cdots i}| > r_{i}(\mathcal{A})\}\neq \varnothing$;

● ($iii$) For any $i\in N_2$, there exists a nonzero element chain from $i$ to $j$ such that $j\in N_1$.

Lemma 4. ^[8,10] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$. If there exists a positive diagonal matrix $X$ such that $\mathcal{A}X^{m-1}$ is an $\mathcal{H}$-tensor, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

2.   Some criteria for judging nonsingular $\mathcal{H}$-tensors

In this section, some new criteria for judging $\mathcal{H}$-tensors are proposed, and those new criteria only depend on the elements of the given tensors.

Theorem 1. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{\lbrack m, n\rbrack}\; (m, n\geq 2)$. $\mathcal{A}$ is an $\mathcal{H}$-tensor, if there exists a number $k = 0, 1, 2, \dots$ such that

Proof. First, let

If $\sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1}}} |a_{ii_{2}\cdots i_{m}}| = 0$, we define $\xi_i = +\infty$. Obviously, it follows from Eq (2.2) that $\xi_i > 0$, $i\in N_2$, and we have $r_{k+1} < r_0 = 1$ by definition of $r_{k+1}$, that is, $1-r_{k+1} > 0$. Hence, there exists a positive number $\varepsilon > 0$, such that

Construct a diagonal matrix $X = diag\{x_1, x_2, \ldots, x_n\}$ and denote $\mathcal{B} = (b_{i_{1}i_{2}\cdots i_{m}}) = \mathcal{A}X^{m-1}$, where

By the inequality of (2.3), we obtain X as a positive diagonal matrix.

Next, we prove the $\sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1}}} |a_{ii_{2}\cdots i_{m}}|\neq 0$ for any $i\in N_2$. Suppose on the contrary that $\sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1}}} |a_{ii_{2}\cdots i_{m}}| = 0$ for any $i\in N_2$; thus, by the inequality of (2.1), we have

which contradicts with $|a_{ii\cdots i}|\leq r_{i}(\mathcal{A})$, $i\in N_2$; hence, $\sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1}}} |a_{ii_{2}\cdots i_{m}}|\neq 0$ for any $i\in N_2$.

Finally, we prove that $\mathcal{B}$ is a strictly diagonally dominant tensor, and we divide it into two cases as follows:

Case 1: For any $i\in N_2$, from $\sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1}}} |a_{ii_{2}\cdots i_{m}}|\neq 0$ and the inequality of (2.1), we have

Case 2: For any $i\in N_1$, we obtain that $|a_{ii\cdots i}| > r_{i}(\mathcal{A})$; then, $|a_{ii\cdots i}|-\sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1}\\\delta_{ii_2\cdots i_m = 0}}}|a_{ii_{2}\cdots i_{m}}| > 0$, and it follows from $r_{k+1}\leq r_{k}$ that

thus, we get

so, we have

From Cases 1 and 2, we obtain that $|b_{ii\cdots i}| > r_{i}(\mathcal{B})$ for all $i\in N$, that is, $\mathcal{B}$ is a strictly diagonally dominant tensor; thus, from Lemmas 1 and 4, $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Theorem 2. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{[m, n]}\; (m, n\geq 2)$. $\mathcal{A}$ is an $\mathcal{H}$-tensor if the following are true:

● ($i$) $\mathcal{A}$ is irreducible.

● ($ii$) There exists $k = 0, 1, 2, \dots$ such that

● ($iii$) For the inequality of ($ii$), strict inequality holds for at least one $i\in N_2$.

Proof. First, let the diagonal matrix $X = diag\{x_1, x_2, \ldots, x_n\}$ and $\mathcal{B} = (b_{i_{1}i_{2}\cdots i_{m}}) = \mathcal{A}X^{m-1}$, where

Obviously, $X$ is the positive diagonal matrix.

Next, we prove that $|b_{ii\dots i}|\geq r_i(\mathcal{B})$ for all $i\in N$, and strict inequality holds for at least one $i\in N$; we have divided it into three cases as follows:

Case 1: For any $i\in N_2$, we obtain

Case 2: For any $i\in N_1$, we obtain

Case 3: From the condition (ⅲ), without loss of generality, we suppose that

similar to the proof for Case 1 of Theorem 2, we obtain that $r_{t}(\mathcal{B}) < |b_{tt\cdots t}|$, $t\in N_2$.

Finally, since $X$ is a positive diagonal matrix and $\mathcal{A}$ is irreducible, $\mathcal{B}$ is also irreducible; thus, by Lemmas 2 and 4, $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Theorem 3. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in {\mathbb{C}}^{[m, n]}\; (m, n\geq 2)$. $\mathcal{A}$ is an $\mathcal{H}$-tensor, if the following are true:

● ($i$) There exists $k = 0, 1, 2, \dots$ such that

● ($ii$) $J\neq \varnothing$, where $J = \left \{j:|a_{jj\cdots j}| > \sum\limits_{\substack{i_{2}\cdots i_{m} \in N^{m-1}\backslash N^{m-1}_1 \\ \delta_{ji_{2}\cdots i_{m}} = 0}} |a_{ji_{2}\cdots i_{m}}|+ \sum\limits_{\substack{i_{2}\cdots i_{m} \in N_1^{m-1}}} r_{k+1} |a_{ji_{2}\cdots i_{m}}|, \; \ j\in N_2 \right\}.$

● ($iii$) For any $i\in (N\backslash J)$, there exists a nonzero element chain from $i$ to $j$ such that $j\in J$.

Proof. First, construct a diagonal matrix $X = diag\{x_1, x_2, \ldots, x_n\}$ and denote $\mathcal{B} = (b_{i_{1}i_{2}\cdots i_{m}}) = \mathcal{A}X^{m-1}$, where

Obviously, $X$ is a positive diagonal matrix.

Second, similar to the proof of Theorem 2, we conclude that $|b_{ii\cdots i}|\geq r_{i}(\mathcal{B})$ for all $i\in N$. From the condition $J\neq \varnothing$, we obtain that there exists at least a $t\in N$ such that $|b_{t t \cdots t}| > r_{t}(\mathcal{B})$. On the other hand, if $|b_{ii\cdots i}| = r_{i}(\mathcal{B})$, then $i\in N\backslash J$, and from the condition that for any $i\in N\backslash J$, $\mathcal{A}$ has a nonzero element chain from $i$ to $j$ such that $j\in J$, we obtain that $\mathcal{B}$ has a nonzero elements chain from $i$ to $j$ with $|b_{jj\cdots j}| > r_{j}(\mathcal{B})$.

Finally, based on the above analysis, we draw a conclusion that $\mathcal{B}$ satisfies the conditions of Lemma 3; hence, by Lemmas 3 and 4, $\mathcal{A}$ is an $\mathcal{H}$-tensor.

3.   Some numerical examples

In this section, based on the new criteria for judging $\mathcal{H}$-tensors in section 2, some numerical examples are presented to illustrate those new criteria.

Example 1. Let us consider the tensor $\mathcal{A} = (a_{i_1i_2i_3}) = [A(1, :, :), A(2, :, :), A(3, :, :)]\in \mathbb{C}^{[3,3]}$, where

Obviously,

so $N_1 = \{1, 2\}$ and $N_2 = \{3\}$. By simple calculation, we obtain

when $k$ = 1, we get

hence, $\mathcal{A}$ satisfies the conditions of Theorem 1 and $k = 1$; it follows from Theorem 1 that $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Example 2. Let us consider the irreducible tensor $\mathcal{A} = (a_{i_1i_2i_3}) = [A(1, :, :), A(2, :, :), A(3, :, :)]\in \mathbb{C}^{[3,3]}$, where

Obviously,

so $N_1 = \{1\}$ and $N_2 = \{2, 3\}$. By simple calculation, we obtain

when $k$ = 0, we get

and

hence, $\mathcal{A}$ satisfies the conditions of Theorem 2 and $k = 0$; it follows from Theorem 2 that $\mathcal{A}$ is an $\mathcal{H}$-tensor.

4.   Application

In this section, based on the new criteria for judging $\mathcal{H}$-tensors in section 2, some new criteria for identifying the positive definiteness of an even-order real symmetric tensor are presented.

From Theorems 1–3, we get the following result.

Theorem 4. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})$ be an even-order real symmetric tensor of order $m$ and $n$ dimensions. If $a_{kk\cdots k} > 0$ for all $k\in N$, $\mathcal{A}$ is symmetric and satisfies one of the following conditions and $\mathcal{A}$ is positive definite:

● ($i$) All conditions of Theorem 1;

● ($ii$) All conditions of Theorem 2;

● ($iii$) All conditions of Theorem 3.

The following example is given to show this result.

Example 3. Consider the following 4th-degree homogeneous polynomial

where $x = (x_1, x_2, x_3)^T$. Then we can obtain a symmetric tensor $\mathcal{A} = (a_{i_1i_2i_3i_4})\in {\mathbb{R}}^{[4,3]}$, where

Obviously,

so $N_1 = \{1, 2\}$ and $N_2 = \{3\}$. By simple calculation, we obtain

Thus, we get

hence, $\mathcal{A}$ satisfies the conditions of Theorem 1 and $k = 0$; thus, it also satisfies the conditions of Theorem 4. Hence, $f(x)$ is positive definite.

5.   Conclusions

In this paper, some new criteria have been proposed for the judgment of $\mathcal{H}$-tensors, which they via an increasing constant $k$ to scale the elements of a given tensor and only depend on elements of the given tensors. As an application, some sufficient conditions of the positive definiteness for even-order real symmetric tensors have been obtained. In addition, some numerical examples have been presented to illustrate those new results.

Acknowledgments

The authors are grateful to the referee for their careful reading of the paper and valuable suggestions and comments. This work is partly supported by the National Natural Science Foundations of China (31600299), Natural Science Basic Research Program of Shaanxi, China (2020JM-622).

Conflict of interest

The authors declare that they have no competing interests.