# Some Properties of the Strong Primitivity of Nonnegative Tensors.

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 m dimension n tensor [mathematical expression not reproducible] over the complex field C is a multidimensional array with all entries

[mathematical expression not reproducible] (1)

A tensor A = ([mathematical expression not reproducible]) is called a nonnegative tensor if all of its entries [mathematical expression not reproducible] 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 A, the following are equivalent [3]:

(1) Let [rho](A) be the spectral radius of A. Then A is irreducible and [rho](A) is greater than any other eigenvalue in modulus.

(2) The only A-invariant nonempty subset of the boundary of the positive cone is {0}.

(3) There exists a natural number r such that [A.sup.r] is positive.

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

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 A be a nonnegative tensor with order m and dimension n, x = [([x.sub.1], [x.sub.2],..., [x.sub.n]).sup.T] [member of] [R.sup.n] a vector, and [x.sup.[r]] = ([x.sup.r.sub.1]. Define the map [T.sub.A] from [R.sup.n] to [mathematical expression not reproducible]. If there exists some positive integer r such that [T.sup.r.sub.A](x) > 0 for all nonnegative nonzero vectors x [member of] [R.sup.n], then A is called primitive and the smallest such integer r is called the primitive degree of A, denoted by y(A).

As in [1], let [mathematical expression not reproducible] be an order m dimension n tensor over the complex field C, [mathematical expression not reproducible], and be a vector [member of] [C.sub.n] whose ith component is defined as follows:

[mathematical expression not reproducible]

Then a number [lambda] [member of] C is called an eigenvalue of A if there exists a nonzero vector x [member of] [C.sub.n] such that

[mathematical expression not reproducible]. (3)

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 [Ax.sup.m-1] can be simply written as A x x.

Definition 2 (see [5]). Let A (and B) be an order m [greater than or equal to] 2 (and k [greater than or equal to] 1), dimension n tensor, respectively. Define the general product A x B (sometimes simplified as AB), to be the following tensor D oforder (m-1)(k-1) + 1 and dimension n:

[mathematical expression not reproducible] (4)

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 [A.sup.k] as the product of k many tensors A.

With the general product, when k =1 and B = x = [([x.sub.1],...,[x.sub.n]).sup.T] [member of] [C.sup.n] is a vector of dimension n, then A x B = A x % is still a vector of dimension n, and for any i [member of] [n], [mathematical expression not reproducible], where [Ax.sup.m-1] is defined in (2). Thus we have [Ax.sup.m-1].

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 A is called essentially positive, if, for any nonnegative nonzero vector x [member of] [R.sup.n], A x x > 0 holds.

Proposition 4 (see [5], Proposition 4.1). Let A be an order m and dimension n nonnegative tensor. Then the following three conditions are equivalent:

(1) For any i, j [member of] [n],[a.sub.ij..j] > 0 holds.

(2) For any j [member of] [n], A x [e.sub.j] > 0 holds (where [e.sub.j] is the j-th column of the identity matrix [I.sub.n]).

(3) For any nonnegative nonzero vector x [member of] [R.sup.n], A x x > 0 holds.

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

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 M(A) of the tensor A is defined as [(M(A)).sub.ij] = [a.sub.ij...j] for i, j [member of] [n].

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 A be a nonnegative tensor with order m and dimension n. Then A is primitive if and only if there exists some positive integer r such that M([A.sup.r]) > 0. Furthermore, the smallest such r is the primitive degree of A, [gamma](A).

On the primitive degree [gamma](A), Shao proposed the following conjecture for further research.

Conjecture 9 (see [5], Conjecture 1). When m is fixed, then there exists some polynomial f(n) on n such that [gamma](A) [less than or equal to] f(n) for all nonnegative primitive tensors oforder m and dimension n.

In the case of m = 2 (A is a matrix), the well-known Wielandt upper bound tells us that we can take f(n) = [(n1).sup.2] + 1. Recently, the authors [7] confirmed Conjecture 9 by proving Theorem 10.

Theorem 10 (see [7], Theorem 1.2). Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree y [gamma(A) [less than or equal to] [(n - l).sup.2] + 1, 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 m [greater than or equal to] 3 is totally different from the case m = 2 (A 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 A be a nonnegative tensor with order m and dimension n. If there exists some positive integer k such that [A.sup.k] > 0 is a positive tensor, then A is called strongly primitive, and the smallest such k is called the strongly primitive degree of A.

Let [mathematical expression not reproducible] be a nonnegative tensor with order m and dimension n. It is clear that if A is strongly primitive, then A is primitive. For convenience, let [eta] (A) be the strongly primitive degree of A. Clearly, [gamma](A) [less than or equal to] [eta](A). In fact, it is obvious that, in the matrix case (m = 2), a nonnegative matrix A is primitive if and only if A is strongly primitive, and [gamma](A) = [eta](A) = exp(A). But in the case m [greater than or equal to] 3 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 j-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 m dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and obtain the characterization of order m dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with n [greater than or equal to] 3 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 [mathematical expression not reproducible] be a nonnegative strongly primitive tensor with order m and dimension n. Then for any [alpha] e [n]m, there exists some i [member of] [n] such that [mathematical expression not reproducible] > 0.

Let k([greater than or equal to] 0), n([greater than or equal to] 2), q([greater than or equal to] 0), r([greater than or equal to] 1) be integers and k = (n - 1)q + r with 1 [less than or equal to] r [less than or equal to] n - 1 when k [greater than or equal to] 1. In [7-9], the authors defined some nonnegative tensors with order m and dimension n as follows:

[mathematical expression not reproducible] (5)

where one has the following: (1)

[mathematical expression not reproducible] (6)

(2) [mathematical expression not reproducible].

(3) [mathematical expression not reproducible].

(4) [mathematical expression not reproducible] 0, except for (1) and (3).

The authors showed the tensors [A.sub.k] (k [greater than or equal to] 0) are primitive, the primitive degree [mathematical expression not reproducible] ([8], Theorem 3.3) for 1 [less than or equal to] k [less than or equal to] [n.sup.2] -3n + 2.

Remark 14. It is clear that, for any [A.sup.k] (0 [less than or equal to] k [less than or equal to] 2 -3n + 2), there exists some [mathematical expression not reproducible]. Thus, for each 0 [less than or equal to] k < n2 -3n + 2, Ak is not a strongly primitive tensor by Proposition 13.

Example 15 ([8], Example 4.17). Let m = n =3, and let A = ([mathematical expression not reproducible]) be a nonnegative tensor with order m and dimension n, where [mathematical expression not reproducible] and other [mathematical expression not reproducible]. Then [eta](A) = 4.

Remark 16. In fact, we can obtain [gamma](A) = [eta](A) = 4 because of [mathematical expression not reproducible], where [mathematical expression not reproducible].

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 A be a nonnegative primitive tensor with order m and dimension [mathematical expression not reproducible]. Then we have

[mathematical expression not reproducible] (7).

Proposition 17 (see [7], Proposition 2.7). Let A be a nonnegative primitive tensor with order m and dimension n and M(A) be the majorization matrix of A. Then we have the following:

(1) For each j [member of] [n], there exists an integer i [member of] [n]\{j} such that [(M(A)).sub.ij] > 0.

(2) There exist some j [member of] [n] and integers u, v with 1 [less than or equal to] u [less than or equal to] v [less than or equal to] n such that [(M(A)).sub.uj] > 0 and [(M(A)).sub.vj] > 0.

Let [mathematical expression not reproducible]. We can see that Proposition 13 is the generalization of result (1) 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 A is a nonnegative strongly primitive tensor, then A must be a nonnegative primitive tensor; thus result (2) of Proposition 17 also holds for nonnegative strongly primitive tensors.

Proposition 18. Let [mathematical expression not reproducible] be a nonnegative strongly primitive tensor with order m and dimension n. Then there exists at least one j [member of] [n] and integers u, v with 1 [less than or equal to] u < v [less than or equal to] n such that [(M(A)).sub.uj] > 0 and [(M(A)).sub.vj] > 0.

Proposition 19. Let [mathematical expression not reproducible] be a nonnegative tensor with order m and dimension n and A [not equal to] J. For given [mathematical expression not reproducible] for any [alpha] [member of] [[n].sup.m- ]1 and any j [member of] [n]\[i}, then A is strongly primitive with [eta] (A) = 2.

Proof. By (7), for any k [member of] [n] and [[alpha].sub.2],..., [[alpha].sub.m] [member of] [[n].sup.m-1], we have

[mathematical expression not reproducible], which implies A is strongly primitive and [eta](A) = 2. ?

Remark 20. From Proposition 19, we can see the following:

(1) There exist at least [mathematical expression not reproducible] 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 i [member of] [n] such that [a.sub.1[alpha]] = 1 > 0 for any [alpha] [member of] [[n].sup.m-]1 and there is exactly one i such that [a.sub.1[alpha]] > 0 for any [alpha] [not equal to] ii...i.

(3) Similarly, we cannot improve the result of Proposition 18 any more by the fact that there is exactly one i [member of] [n] such that [(M(A)).sup.ui] > 0 for any u [member of] [n] and for any other j [member of] [n]\{i}, there exists only i [member of] [n] such that [(M(A)).sub.ij] > 0.

(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 [mathematical expression not reproducible] be a nonnegative strongly primitive tensor with order m and dimension n. Then for any i [member of] [n], there exists some [alpha] [member of] [[n].sup.m-1] such that [a.sub.i[alpha]] = 0 > 0.

Proof. Since A is strongly primitive, there exists some k >0 such that [A.sup.k] > 0 by Definition 1. Assume that there exists some i [member of] [n] such that [a.sub.1[alpha]] = 0 for any [alpha] [member of] [[n].sup.m-1]. Then by (7), we have

[mathematical expression not reproducible] (8)

Remark 22. Let [mathematical expression not reproducible] be a nonnegative tensor with order m and dimension n. For given i [member of] [n], we take [a.sub.1[alpha]] = [a.sub.jii.ii] = 1 for any [alpha] [member of] [[n].sup.m-1] and any j [member of] [n]\{i} and any other entry [mathematical expression not reproducible]. Then A is strongly primitive with [eta](A) = 2 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 A be a nonnegative strongly primitive tensor and k = [eta](A). Then, for any integer t > k >0, we have [A.sup.t] > 0.

Proof. It is clear that [A.sup.k] > 0 by k = [eta](A). We only need to show [A.sup.k+1] > 0; say, for any i [member of] [n] and any [[alpha].sub.2],..., [[alpha].sub.2] [member of] [mathematical expression not reproducible]

By Proposition 21, there exists some [alpha] = [j.sub.2][j.sub.3]... [j.sub.m] [member of] [[n].sup.m-1] such that [mathematical expression not reproducible]. By [A.sup.k] > 0 we have [mathematical expression not reproducible] > 0 then by (7) we have

[mathematical expression not reproducible].

Proposition 24. Let A be a nonnegative tensor with order m and dimension n and t be a positive integer. Then A is strongly primitive if and only if [A.sup.t] is strongly primitive.

Proof. Firstly, the sufficiency is obvious. Now we show the necessity. Let k = [eta](A). Then [A.sup.k] > 0 by A is strongly primitive. Let s be a positive integer such that st [greater than or equal to] k; then A[s.sup.t] > 0 by Proposition 23. Thus [([A.sup.t]).sup.s] = [A.sup.st] > 0, which implies [A.sup.t] is strongly primitive.

3. A Characterization of the (Strongly) Primitive Tensor with Order m and Dimension 2

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

Lemma 25 (see [5], Corollary 4.1). Let A be a nonnegative tensor with order m and dimension n. If M(A) is primitive, then A is also primitive and in this case we have [gamma](A) [less than or equal to] [gamma](M(A)) [less than or equal to] [(n- l).sup.2] + 1.

Theorem 26. Let A be a nonnegative tensor with order m and dimension n = 2. Then A is primitive if and only if M(A) 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:

[mathematical expression not reproducible] (10)

Let A be primitive. Then [gamma](A) [less than or equal to] 2 by Theorem 10 and M([A.sup.2]) >0 by Proposition 8. Now we assume that M(A) is not primitive; we will show A is also not primitive.

It is not difficult to find that

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

In (12), we note that [i.sub.2][i.sub.3]... [i.sub.m] [not equal to] 22... 2, which implies that there exists at least one entry, say, [i.sub.s] = l, where 2 [less than or equal to] s [less than or equal to] m; then [mathematical expression not reproducible].

Similarly, in (13), we note that [i.sub.2][i.sub.3]... [i.sub.m] [not equal to] 11...1, which implies that there exists at least one entry, say, [i.sub.s] = 2, where [mathematical expression not reproducible].

Thus, by (12), (13), and the above arguments, we have

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

Since M(A) is not primitive; by (10), we can complete the proof by the following two cases.

Case 1. [mathematical expression not reproducible].

Subcase 1.1. [M(A).sub.12] = 0.

Then [a.sub.12...2] = 0. By (14), we have [M([A.sup.2]).sub.12] = [([A.sup.2]).sub.12...2] = 0, which implies [A.sup.2] is not essentially positive.

Subcase 1.2.M(A)21 = 0.

Then [a.sub.21...1] = 0. By (15), wehave [M([A.sup.2]).sub.21] = [([A.sup.2]).sub.21...]1 = 0, which implies [A.sup.2] is not essentially positive.

Case 2.[mathematical expression not reproducible].

Then we have [M(A).sub.11] = [M(A).sub.22] = 0; that is, [a.sub.11...1] = [a.sub.22...2] = 0; by (15) we have [M([A.sup.2]).sub.12] = ([[A.sup.2]).sub.12...2] = 0, which implies [A.sup.2] 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 n [greater than or equal to] 3 or not. The following Example 27 shows that the necessity of Theorem 26 is false with n [greater than or equal to] 3.

Example 27. Let [mathematical expression not reproducible] be a nonnegative tensor of order m and dimension n [greater than or equal to] 3, where

[mathematical expression not reproducible](16)

Then A is (strongly) primitive, but M(A) is not primitive.

Proof. By direct calculation and Definition 2, we know that [A.sup.2] is the tensor of order [(m - 1).sup.2] + 1 and dimension n, and for any 1 [less than or equal to] i [less than or equal to] n, we have

[mathematical expression not reproducible] (17)

Obviously, [A.sup.2] is positive; then A is strongly primitive with [eta](A) = 2 and thus A is primitive with [gamma](A) = 2.

On the other hand, by the definition of A, we have

[mathematical expression not reproducible] (18)

Since the associated digraph of M(A) is not strongly connected, thus M(A) is not primitive.

Next, we will study the strongly primitive degree of order m and dimension 2 tensors. Firstly, we discuss an example with order m = 5 and dimension n = 2 tensor as follows.

Definition 28 (see [10]). Let A be a tensor with order m and dimension n. The i-th slice of A, denoted by A[i], is the subtensor of A with order m - 1 and dimension n such that [mathematical expression not reproducible].

Example 29. Let [mathematical expression not reproducible] be a nonnegative tensor with order m = 5 and dimension n = 2, where [a.sub.12122] = [a.sub.21121] = 0 and other [mathematical expression not reproducible]. Then there exists at least one zero element in each slice of [A.sup.2].

Proof. Let [[alpha].sub.1] = 2122, [[alpha].sub.2] = 1121, and denote [[beta].sub.2] = [[beta].sub.4] = [[beta].sub.5] = [[alpha].sub.1], [[beta].sub.3] = [[alpha].sub.2]. Then we have

[mathematical expression not reproducible] (19)

Similarly, we let [[gamma].sub.2] = [[gamma].sub.3] = [[gamma].sub.5] = [[alpha].sub.2] and [[gamma].sub.4] = [[alpha]sub.1], we can show [mathematical expression not reproducible], and we omit it.

Combining the above arguments, we know there exists at least one zero element in each slice of [A.sup.2] by [mathematical expression not reproducible] = 0.

Similarly, the result of Example 29 can be generalized to any nonnegative tensor with order m and dimension n = 2.

Lemma 30. Let [mathematical expression not reproducible] be a nonnegative tensor with order m and dimension n = 2. If there exist [mathematical expression not reproducible] such that [mathematical expression not reproducible]. Then

[mathematical expression not reproducible] (20)

Proof. We first show [mathematical expression not reproducible]. For any 2 [less than or equal to] t [less than or equal to] m, [j.sub.t] [member of] {1,2}, we denote [[bar.j].sub.t] [member of] {1,2}\{[j.sub.t]}. Then we have [mathematical expression not reproducible]

[mathematical expression not reproducible] (21)

Similarly, for any 2 [less than or equal to] t [less than or equal to] m, [k.sub.t] {1,2}, we denote [mathematical expression not reproducible]. Then we have [mathematical expression not reproducible] and we can show [mathematical expression not reproducible] by

[mathematical expression not reproducible] (22)

and similar process of the above arguments. Thus we complete the proof of (20).

Theorem 31. Let [mathematical expression not reproducible] be a nonnegative tensor with order m and dimension n =2. If there exist [mathematical expression not reproducible] such that [mathematical expression not reproducible], then A is not strongly primitive.

Proof. Now we show that there exists at least one zero element in each slice of [A.sup.r] by induction on r([greater than or equal to] 2).

Firstly, by Lemma 30, we know there exists at least one zero element in each slice of [A.sup.2]. Now we assume that there exists at least one zero element in each slice of [A.sup.r-1]; say, there exist [mathematical expression not reproducible] such that [mathematical expression not reproducible]. Then by (7) and the similar proof of Lemma 30, we have

[mathematical expression not reproducible] (23)

and

[mathematical expression not reproducible] (24)

By (23) and (24), we obtain that there exists at least one zero element in each slice of [A.sup.r], and thus we complete the proof.

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

Theorem 32. Let [mathematical expression not reproducible] be a nonnegative tensor with order m and dimension n = 2. Then 1

(1) A is strongly primitive if and only if one of the following holds:

(a) A = J;

(b) [mathematical expression not reproducible];

(c) [mathematical expression not reproducible];

(2) if A is strongly primitive, then [eta](A) [less than or equal to] 2.

Proof. Firstly, we show that (1) is sufficient. It is easy to see that A = J is strongly primitive with [eta] (J) = 1, and if A satisfies (b) or (c), A is strongly primitive with [eta](A) = 2 by Proposition 19 immediately.

Now we show the necessity of (1); that is, if A does not satisfy the conditions of (a), (b), or (c), then we will show that A is not strongly primitive. We complete the proof by the following three cases.

Case 1. [a.sub.[alpha]] =1 for any a e [[2].sup.m-1] and [a.sub.211 ... 1] = 0.

It is not difficult to find that [mathematical expression not reproducible]. Then A is not primitive by Theorem 26, and thus A is not strongly primitive.

Case 2. [a.sub.2a] = 1 for any [alpha] [member of] [[2].sup.m-1] and [a.sub.122...2] = 0.

Similarly, we can find that M(A) = (11). Then A is not primitive by Theorem 26, and thus A is not strongly primitive.

Case 3. There is at least one zero element in each slice of A.

Then there exist [mathematical expression not reproducible]. Thus A is not strongly primitive by Theorem 31.

(2) If A is strongly primitive, by Definition 11 and the proof of (1), we obtain [eta] (A) [less than or equal to] 2 immediately.

Remark 33. By Theorem 32, we can see that the strongly primitive degree [eta](A) of an nonnegative tensor with order m and dimension n = 2 is irrelevant to its order

4. Some Properties and Problems of Order m Dimension n([greater than or equal to] 3) Strongly Primitive Tensors

In this section, we will study some properties of the strongly primitive tensors with order m and dimension n [greater than or equal to] 3 and propose some questions for further research.

Proposition 34. Let [mathematical expression not reproducible] be a nonnegative tensor with order m and dimension n. Let [mathematical expression not reproducible], then there exist [mathematical expression not reproducible].

Proof. For each [mathematical expression not reproducible].

[mathematical expression not reproducible] (25)

We note that k [member of] [n]\{i} which means there are n-1 zero elements in i-th slice of [A.sup.2]; thus we complete the proof by I [member of] [n].

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

Theorem 35. Let [mathematical expression not reproducible] be a nonnegative tensor with order m and dimension n. Let [mathematical expression not reproducible] and any s [member of] [n]\{i}, then A is not strongly primitive.

Proof. Now we show that there exist [mathematical expression not reproducible] induction on r([greater than or equal to]2); say, there exist at least n-1 zero elements in each slice of [A.sup.r] and thus A is not strongly primitive.

Firstly, by Proposition 34, we know there exist [mathematical expression not reproducible] such that [mathematical expression not reproducible] for any i [member of] [n] and any k e [n]\{1}; say, there exist at least n-1 zero elements in each slice of [A.sup.2]. Now we assume that there exist [mathematical expression not reproducible] and any k [member of] [n]\{i}; say, there exist at least n-1 zero elements in each slice of [A.sup.r-1].

Let [mathematical expression not reproducible] for any s [member of] [n] and 2 [less than or equal to] t [less than or equal to] m; then [mathematical expression not reproducible] = 0 for any i [member of] [n] and any k [member of] [n]\{i}. Let [mathematical expression not reproducible] for any k [member of] [n]. Now we show [mathematical expression not reproducible] for any i [member of] [n] and any k [member of] [n]\{i}.

By (7) and similar proof of Proposition 34, we have

[mathematical expression not reproducible] (26)

and then we complete the proof.

Proposition 36. Let [mathematical expression not reproducible] be a non-negative tensor with order m and dimension n, and M(A) be the majorization matrix of A. If there exist i, j [member of] [n], such that [(M(A)).sub.ij] > 0,[(M(A)).sub.vi]: = 0 for any u [member of] [n]\{i} and [(M(A)).sub.ji] > 0, [(M(A)).sub.vi] = 0 for any v [member of] [n]\{j}, then A is not primitive, and thus A is not strongly primitive.

Proof. Firstly, we show the following assertion:

If k is odd, then [(M([A.sup.k])).sub.ij] > 0, (M([A.sup.k]))jt > 0, [(M(Ak)).sub.uj] = 0 for any u [member of] [n],\{i} [(M([A.sup.k])).sub.vi] = 0 for any v [member of] [n]/{j}.

If k is even, then [mathematical expression not reproducible] for any v [member of] [n]\{i}.

When k = 1, the above result holds which is obvious. When k = 2, by Definition 7 and (7), we have

[mathematical expression not reproducible] (27)

And for any u [member of] [n]|{i}, we have

[mathematical expression not reproducible] (28)

Similarly, we can show [(M([A.sup.2])).sub.jj] > 0 and [(M([A.sup.2])).sub.vj] = 0 for any v [member of] [n]\{j}.

Now we assume that, for any k, the above assertion holds. Then, for k + 1, we consider the following two cases.

Case 1. k is odd.

Then, by (7), we have

[mathematical expression not reproducible] (29)

and, for any u e [n]\{i}, we have

[mathematical expression not reproducible] (30)

Similarly, we can show [mathematical expression not reproducible].

Case 2. k is even.

By (7) and similar proof of Case 1, we can show [mathematical expression not reproducible] for any u [member of] [n]\{i}, and (M[(A.sup.k+1])).sub.vi] = 0 for any v e [n]\{j}.

By Proposition 8 and the above assertion, we know A is not primitive, and thus A is not strongly primitive.

Let [mathematical expression not reproducible] be a nonnegative strongly primitive tensor with order m and dimension n. When n = 2, we know n(A) [less than or equal to] 2 by Theorem 32. When n [greater than or equal to] 3, we do not know the value or bound of n(A). Even n = 3, we do not find out all strongly primitive tensors. Thus we think it is not easy to obtain the value or bound of n(A). Based on the computation of the case n = 3, we propose the following problem for further research.

Question 37 Let n [greater than or equal to] 3, [mathematical expression not reproducible] be a nonnegative strongly primitive tensor with order m and dimension n. Then n(A) < [(n- 1).sup.2] + 1.

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).

References

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[7] P. Yuan, Z. He, and L. You, "A conjecture on the primitive degree of tensors," Linear Algebra and Its Applications,vol. 450, pp. 175-185, 2014.

[8] P. Yuan, Z. He, and L. You, "Further results and some open problems on the primitive degree of nonnegative tensors," Linear Algebra and Its Applications, vol. 480, pp. 72-92, 2015.

[9] Z. He, P. Yuan, and L. You, "On the exponent set of nonnegative primitive tensors," Linear Algebra and Its Applications, vol. 465, pp. 376-390, 2015.

[10] J.-Y. Shao, H.-Y. Shan, and L. Zhang, "On some properties of the determinants oftensors," Linear Algebra and Its Applications, vol. 439, no. 10, pp. 3057-3069, 2013.

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Lihua You [ID], Yafei Chen, and Pingzhi Yuan [ID]

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Correspondence should be addressed to Lihua You; ylhua@scnu.edu.cn and Pingzhi Yuan; yuanpz@scnu.edu.cn

Received 25 January 2018; Accepted 3 July 2018; Published 7 August 2018