Some Properties of the Strong Primitivity of Nonnegative Tensors

Abstract

We show that an order dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we obtain the characterization of order dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with and propose some problems for further research.

1. Introduction

In recent years, the study of tensors and the spectra of tensors (and hypergraphs) with their various applications has attracted extensive attention and interest, since the work of L. Qi ([1]) and L.H. Lim ([2]) in 2005.

As is in [1], an order dimension tensor over the complex field is a multidimensional array with all entries

A tensor is called a nonnegative tensor if all of its entries are nonnegative. Clearly, adjacency tensors and signless Laplacian tensors are nonnegative.

In the theory of nonnegative matrices, the notion of primitivity plays an important role in the convergence of the Collatz method. For a nonnegative matrix , the following are equivalent [3]:

Let be the spectral radius of . Then is irreducible and is greater than any other eigenvalue in modulus.

The only -invariant nonempty subset of the boundary of the positive cone is .

There exists a natural number such that is positive.

Matrices which satisfy any of the above conditions are called primitive. The least such such that is positive is called the primitive exponent (or simply, exponent) of and is denoted by .

In [4], K.C. Chang et al. defined the primitivity of nonnegative tensors (as Definition 1), extended the theory of nonnegative matrices to nonnegative tensors, and proved the convergence of the NQZ method which is an extension of the Collatz method and can be used to find the largest eigenvalue of any nonnegative irreducible tensor.

Definition 1 (see [4]). Let be a nonnegative tensor with order and dimension , a vector, and . Define the map from to as . If there exists some positive integer such that for all nonnegative nonzero vectors , then is called primitive and the smallest such integer is called the primitive degree of , denoted by .

As in [1], let be an order dimension tensor over the complex field , , , and be a vector whose th component is defined as follows:Then a number is called an eigenvalue of if there exists a nonzero vector such that

Recently, Shao [5] defined the general product of two n-dimensional tensors as follows, and one of the applications of the tensor product is that can be simply written as

Definition 2 (see [5]). Let (and ) be an order (and ), dimension tensor, respectively. Define the general product (sometimes simplified as ), to be the following tensor of order and dimension :

The tensor product is a generalization of the usual matrix product and satisfies a very useful property: the associative law ([5], Theorem 1.1). By the associative law, we can define as the product of many tensors .

With the general product, when and is a vector of dimension , then is still a vector of dimension , and for any , , where is defined in (2). Thus we have

In order to study eigenvalue, Pearson defined “essentially positive” tensors as Definition 3. By the general product of tensors, Shao obtained Proposition 4 and Definition 5 which is equivalent to Definition 3.

Definition 3 (see [6], Definition 3.1). A nonnegative tensor is called essentially positive, if, for any nonnegative nonzero vector holds.

Proposition 4 (see [5], Proposition 4.1). Let be an order and dimension nonnegative tensor. Then the following three conditions are equivalent:
(1) For any holds.
(2) For any holds (where is the -th column of the identity matrix ).
(3) For any nonnegative nonzero vector holds.

Definition 5 (see [5], Definition 4.1). A nonnegative tensor is called essentially positive, if it satisfies one of the three conditions in Proposition 4.

Based on the above arguments and the zero patterns defined by Shao in [5], Shao showed a characterization of primitive tensors and defined the primitive degree as follows.

Proposition 6 (see [5], Theorem 4.1). A nonnegative tensor is primitive if and only if there exists some positive integer such that is essentially positive. Furthermore, the smallest such is the primitive degree of , .

The concept of the majorization matrix of a tensor introduced by Pearson is very useful.

Definition 7 (see [6], Definition 2.1). The majorization matrix of the tensor is defined as for .

By Definition 5, Proposition 6, and Definition 7, the following characterization of the primitive tensors was easily obtained.

Proposition 8 (see [7], Remark 2.6). Let be a nonnegative tensor with order and dimension . Then is primitive if and only if there exists some positive integer such that Furthermore, the smallest such is the primitive degree of , .

On the primitive degree , Shao proposed the following conjecture for further research.

Conjecture 9 (see [5], Conjecture 1). When is fixed, then there exists some polynomial on such that for all nonnegative primitive tensors of order and dimension .

In the case of ( is a matrix), the well-known Wielandt upper bound tells us that we can take Recently, the authors [7] confirmed Conjecture 9 by proving Theorem 10.

Theorem 10 (see [7], Theorem 1.2). Let be a nonnegative primitive tensor with order and dimension . Then its primitive degree , and the upper bound is tight.

They also showed that there are no gaps in the tensor case in [8], which implies that the result of the case is totally different from the case ( is a matrix). In [5], Shao also proposed the concept of strongly primitive tensor for further research.

Definition 11 (see [5], Definition 4.3). Let be a nonnegative tensor with order and dimension . If there exists some positive integer such that is a positive tensor, then is called strongly primitive, and the smallest such is called the strongly primitive degree of .

Let be a nonnegative tensor with order and dimension . It is clear that if is strongly primitive, then is primitive. For convenience, let be the strongly primitive degree of . Clearly, . In fact, it is obvious that, in the matrix case , a nonnegative matrix is primitive if and only if is strongly primitive, and . But in the case Shao gave an example to show that these two concepts are not equivalent. In [8], the authors proposed the following question.

Question 12 ([8], Question 4.18). Can we define and study the strongly primitive degree, the strongly primitive degree set, the -strongly primitive degree of strongly primitive tensors and so on?

Based on Question 12, we study primitive tensors and strongly primitive tensors in this paper, show that an order dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and obtain the characterization of order dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with and propose some problems for further research.

2. Preliminaries

In [8], the authors obtained the following Proposition 13 and gave Example 15 by computing the strongly primitive degree.

Proposition 13 ([8], Proposition 4.16). Let be a nonnegative strongly primitive tensor with order and dimension . Then for any , there exists some such that .

Let , , , be integers and with when . In [7–9], the authors defined some nonnegative tensors with order and dimension as follows: where one has the following:

(1)

(2) , if for any .

(3) , if and with .

(4) , except for and .

The authors showed the tensors are primitive, the primitive degree ([7]) and ([8], Theorem 3.3) for .

Remark 14. It is clear that, for any , there exists some , for any , . Thus, for each , is not a strongly primitive tensor by Proposition 13.

Example 15 ([8], Example 4.17). Let , and let be a nonnegative tensor with order and dimension , where and other . Then

Remark 16. In fact, we can obtain because of , where .

In the computation of Example 15, we note that the following equation is useful and will be used repeatedly. It is not difficult to obtain the equation by the general product of two n-dimensional tensors which is defined in Definition 1.2 in [5].

Let be a nonnegative primitive tensor with order and dimension , Then we have

Proposition 17 (see [7], Proposition 2.7). Let be a nonnegative primitive tensor with order and dimension and be the majorization matrix of . Then we have the following:
(1) For each , there exists an integer such that .
(2) There exist some and integers with such that and .

Let ; then . We can see that Proposition 13 is the generalization of result of Proposition 17 from a primitive tensor to a strongly primitive tensor. We note that Proposition 17 played an important role in [7], and if is a nonnegative strongly primitive tensor, then must be a nonnegative primitive tensor; thus result of Proposition 17 also holds for nonnegative strongly primitive tensors.

Proposition 18. Let be a nonnegative strongly primitive tensor with order and dimension . Then there exists at least one and integers with such that and .

Proposition 19. Let be a nonnegative tensor with order and dimension and . For given , if for any and any , then is strongly primitive with .

Proof. By (7), for any and , we have
, which implies is strongly primitive and .

Remark 20. From Proposition 19, we can see the following:
(1) There exist at least strongly primitive tensors such that their strongly primitive degree is equal to 2.
(2) We cannot improve the result of Proposition 13 any more by the fact that there exists such that for any and there is exactly one such that for any .
(3) Similarly, we cannot improve the result of Proposition 18 any more by the fact that there is exactly one such that for any and for any other , there exists only such that .
(4) What is more, combining the above arguments, we know whether a nonnegative tensor is a nonnegative strongly primitive tensor or not, and the values of the strongly primitive degree of a nonnegative strongly primitive tensor do not depend on the number of nonzero entries but the positions of the nonzero entries.

Proposition 21. Let be a nonnegative strongly primitive tensor with order and dimension . Then for any , there exists some such that .

Proof. Since is strongly primitive, there exists some such that by Definition 1. Assume that there exists some such that for any . Then by (7), we have which leads to a contraction.

Remark 22. Let be a nonnegative tensor with order and dimension . For given , we take for any and any and any other entry . Then is strongly primitive with by Proposition 19. This implies that we cannot improve the result of Proposition 21 anymore, and it indicates the importance of the positions of the nonzero entries again.

Proposition 23. Let be a nonnegative strongly primitive tensor and . Then, for any integer , we have .

Proof. It is clear that by . We only need to show ; say, for any and any , we show .
By Proposition 21, there exists some such that . By we have ; then by (7), we have

Proposition 24. Let be a nonnegative tensor with order and dimension and be a positive integer. Then is strongly primitive if and only if is strongly primitive.

Proof. Firstly, the sufficiency is obvious. Now we show the necessity. Let . Then by is strongly primitive. Let be a positive integer such that ; then by Proposition 23. Thus , which implies is strongly primitive.

3. A Characterization of the (Strongly) Primitive Tensor with Order and Dimension

In this section, we study primitive tensors and strongly primitive tensors in this paper, show that an order dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and obtain the characterization of order dimension 2 strongly primitive tensors and the bound of the strongly primitive degree.

Lemma 25 (see [5], Corollary 4.1). Let be a nonnegative tensor with order and dimension . If is primitive, then is also primitive and in this case we have .

Theorem 26. Let be a nonnegative tensor with order and dimension . Then is primitive if and only if is primitive.

Proof. Firstly, the sufficiency is obvious by Lemma 25. Now we only show the necessity. Clearly, all primitive (0,1) matrices of order 2 are listed as follows:Let be primitive. Then by Theorem 10 and by Proposition 8. Now we assume that is not primitive; we will show is also not primitive.
It is not difficult to find thatIn (12), we note that , which implies that there exists at least one entry, say, , where ; then
Similarly, in (13), we note that , which implies that there exists at least one entry, say, , where ; then
Thus, by (12), (13), and the above arguments, we haveSince is not primitive; by (10), we can complete the proof by the following two cases.
Case 1. .
Subcase 1.1..
Then . By (14), we have , which implies is not essentially positive.
Subcase 1.2..
Then . By (15), we have , which implies is not essentially positive.
Case 2. .
Then we have ; that is, ; by (15) we have , which implies is not essentially positive.
Based on the above two cases and Proposition 6, we complete the proof of the necessity.

A nature question is whether the result of Theorem 26 is true for or not. The following Example 27 shows that the necessity of Theorem 26 is false with .

Example 27. Let be a nonnegative tensor of order and dimension , where Then is (strongly) primitive, but is not primitive.

Proof. By direct calculation and Definition 2, we know that is the tensor of order and dimension , and for any , we have Obviously, is positive; then is strongly primitive with and thus is primitive with .
On the other hand, by the definition of , we haveSince the associated digraph of is not strongly connected, thus is not primitive.

Next, we will study the strongly primitive degree of order and dimension 2 tensors. Firstly, we discuss an example with order and dimension tensor as follows.

Definition 28 (see [10]). Let be a tensor with order and dimension . The -th slice of , denoted by , is the subtensor of with order and dimension such that .

Example 29. Let be a nonnegative tensor with order and dimension , where and other . Then there exists at least one zero element in each slice of .

Proof. Let , , and denote , . Then we haveSimilarly, we let and , we can show , and we omit it.
Combining the above arguments, we know there exists at least one zero element in each slice of by .

Similarly, the result of Example 29 can be generalized to any nonnegative tensor with order and dimension .

Lemma 30. Let be a nonnegative tensor with order and dimension . If there exist and such that , for any , let and Then

Proof. We first show . For any , , we denote . Then we have by , andSimilarly, for any , , we denote . Then we have and we can show by and similar process of the above arguments. Thus we complete the proof of (20).

Theorem 31. Let be a nonnegative tensor with order and dimension . If there exist and such that , then is not strongly primitive.

Proof. Now we show that there exists at least one zero element in each slice of by induction on .
Firstly, by Lemma 30, we know there exists at least one zero element in each slice of . Now we assume that there exists at least one zero element in each slice of ; say, there exist such that . For any , let and Then by (7) and the similar proof of Lemma 30, we haveandBy (23) and (24), we obtain that there exists at least one zero element in each slice of , and thus we complete the proof.

Now we give the characterization of the strongly primitive tensor with order and dimension 2.

Theorem 32. Let be a nonnegative tensor with order and dimension . Then(1) is strongly primitive if and only if one of the following holds:(a) ;(b) and ;(c) and ;(2) if is strongly primitive, then .

Proof. Firstly, we show that (1) is sufficient. It is easy to see that is strongly primitive with , and if satisfies (b) or (c), is strongly primitive with by Proposition 19 immediately.
Now we show the necessity of (1); that is, if does not satisfy the conditions of (a), (b), or (c), then we will show that is not strongly primitive. We complete the proof by the following three cases.
Case 1. for any and .
It is not difficult to find that . Then is not primitive by Theorem 26, and thus is not strongly primitive.
Case 2. for any and .
Similarly, we can find that . Then is not primitive by Theorem 26, and thus is not strongly primitive.
Case 3. There is at least one zero element in each slice of .
Then there exist and such that . Thus is not strongly primitive by Theorem 31.
(2) If is strongly primitive, by Definition 11 and the proof of , we obtain immediately.

Remark 33. By Theorem 32, we can see that the strongly primitive degree of an nonnegative tensor with order and dimension is irrelevant to its order .

4. Some Properties and Problems of Order Dimension Strongly Primitive Tensors

In this section, we will study some properties of the strongly primitive tensors with order and dimension and propose some questions for further research.

Proposition 34. Let be a nonnegative tensor with order and dimension . Let , , , and . If for any and any , then there exist such that for any and any .

Proof. For each , let for any and ; then for any by for any and any . Let for any . Now we show for any .We note that which means there are zero elements in -th slice of ; thus we complete the proof by .

We note that Proposition 34 is the generalization of Lemma 30; now we will obtain the generalization of Theorem 31.

Theorem 35. Let be a nonnegative tensor with order and dimension . Let , , , and . If for any and any , then is not strongly primitive.

Proof. Now we show that there exist such that for any and any by induction on ; say, there exist at least zero elements in each slice of and thus is not strongly primitive.
Firstly, by Proposition 34, we know there exist such that for any and any ; say, there exist at least zero elements in each slice of . Now we assume that there exist such that for any and any ; say, there exist at least zero elements in each slice of .
Let for any and ; then for any by for any and any . Let for any . Now we show for any and any .
By (7) and similar proof of Proposition 34, we have and then we complete the proof.

Proposition 36. Let be a nonnegative tensor with order and dimension , and be the majorization matrix of . If there exist , such that for any and for any , then is not primitive, and thus is not strongly primitive.

Proof. Firstly, we show the following assertion:
If   is odd, then  ,  ,   for any  ,   for any  .
If   is even, then  ,  ,   for any  ,   for any  .
When , the above result holds which is obvious. When , by Definition 7 and (7), we haveand for any , we haveSimilarly, we can show and for any .
Now we assume that, for any , the above assertion holds. Then, for , we consider the following two cases.
Case 1. is odd.
Then, by (7), we haveand, for any , we haveSimilarly, we can show and for any .
Case 2. is even.
By (7) and similar proof of Case 1, we can show , , for any , and for any .
By Proposition 8 and the above assertion, we know is not primitive, and thus is not strongly primitive.

Let be a nonnegative strongly primitive tensor with order and dimension . When , we know by Theorem 32. When , we do not know the value or bound of . Even , we do not find out all strongly primitive tensors. Thus we think it is not easy to obtain the value or bound of . Based on the computation of the case , we propose the following problem for further research.

Question 37. Let , be a nonnegative strongly primitive tensor with order and dimension . Then

In [7, 9], the authors gave some algebraic characterizations of a nonnegative primitive tensor, and in [11] the authors showed that a nonnegative tensor is primitive if and only if the greatest common divisor of all the cycles in the associated directed hypergraph is equal to 1. It is natural for us to consider the following.

Question 38. Study the algebraic or graphic characterization of a nonnegative strongly primitive tensor.

We are sure the above two questions are interesting and not easy.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

L. You’s research is supported by the National Natural Science Foundation of China (Grant no. 11571123) and the Guangdong Provincial Natural Science Foundation (Grant no. 2015A030313377). P. Yuan’s research is supported by the NSF of China (Grant no. 11671153). Y. Chen’s research is supported by the Scientific Research Foundation of Graduate School of South China Normal University (Grant no. 2015lkxm19).