Some Properties of the Strong Primitivity of Nonnegative Tensors

Copyright © 2018 Lihua You et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We show that an orderm dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we 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 ≥ 3 and propose some problems for further research.


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.
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]: (1) Let () be the spectral radius of .Then  is irreducible and () is greater than any other eigenvalue in modulus.
(2) The only -invariant nonempty subset of the boundary of the positive cone is {0}.
(3) 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 exp().
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  and dimension ,  = ( 1 ,  2 , . . .,   )  ∈ R  a vector, and  [] = (  1 ,   2 , . . .,    )  .Define the map ] .If there exists some positive integer  such that   A () > 0 for all nonnegative nonzero vectors  ∈ R  , then A is called primitive and the smallest such integer  is called the primitive degree of A, denoted by (A).
As in [1], let A = (  1  2 ...  ) 1≤  ≤ (=1,...,) be an order  dimension  tensor over the complex field C,  = ( 1 , . . .,   )  ∈ C  ,  [] = (  1 ,   2 , . . .,    ), and A −1 be a vector ∈ C  whose th component is defined as follows: Then a number  ∈ C is called an eigenvalue of A if there exists a nonzero vector  ∈ C  such that Definition 2 (see [5]).Let A (and B) be an order  ≥ 2 (and  ≥ 1), dimension  tensor, respectively.Define the general product A ⋅ B (sometimes simplified as AB), to be the following tensor D of order (−1)(−1)+1 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 A  as the product of  many tensors A.
With the general product, when  = 1 and B =  = ( 1 , . . .,   )  ∈ C  is a vector of dimension , then A⋅B = A⋅ is still a vector of dimension , and for any  ∈ 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  ∈ R  , A ⋅  > 0 holds.Proposition 4 (see [5], Proposition 4.1).Let A be an order  and dimension  nonnegative tensor.Then the following three conditions are equivalent: (1) For any ,  ∈ [],  ... > 0 holds.
(2) For any  ∈ [], A ⋅   > 0 holds (where   is the -th column of the identity matrix   ).
(3) For any nonnegative nonzero vector  ∈ R  , A ⋅  > 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  such that A  is essentially positive.Furthermore, the smallest such  is the primitive degree of A, (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 (A) of the tensor A is defined as ((A))  =  ... 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 A be a nonnegative tensor with order  and dimension .Then A is primitive if and only if there exists some positive integer  such that (A  ) > 0. Furthermore, the smallest such  is the primitive degree of A, (A).
On the primitive degree (A), 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 (A) ≤ () for all nonnegative primitive tensors of order  and dimension .
In the case of  = 2 (A is a matrix), the well-known Wielandt upper bound tells us that we can take () = ( − 1) 2 + 1.Recently, the authors [7] confirmed Conjecture 9 by proving Theorem 10.
They also showed that there are no gaps in the tensor case in [8], which implies that the result of the case  ≥ 3 is totally different from the case  = 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  and dimension .If there exists some positive integer  such that A  > 0 is a positive tensor, then A is called strongly primitive, and the smallest such  is called the strongly primitive degree of A.
Let A = (  1  2 ...  ) be a nonnegative tensor with order  and dimension .It is clear that if A is strongly primitive, then A is primitive.For convenience, let (A) be the strongly primitive degree of A. Clearly, (A) ≤ (A).In fact, it is obvious that, in the matrix case ( = 2), a nonnegative matrix  is primitive if and only if  is strongly primitive, and () = () = exp().But in the case  ≥ 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 -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  ≥ 3 and propose some problems for further research.

Preliminaries
In [8], the authors obtained the following Proposition 13 and gave Example 15 by computing the strongly primitive degree.
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  =  . . . ∈ [] −1 ; then (A)  =   .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.
(2) We cannot improve the result of Proposition 13 any more by the fact that there exists  ∈ [] such that   = 1 > 0 for any  ∈ [] −1 and there is exactly one  such that   > 0 for any  ̸ =  . . .. (3) Similarly, we cannot improve the result of Proposition 18 any more by the fact that there is exactly one  ∈ [] such that ((A))  > 0 for any  ∈ [] and for any other  ∈ []\{}, there exists only  ∈ [] such that ((A))  > 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.
Proof.Since A is strongly primitive, there exists some  > 0 such that A  > 0 by Definition 1. Assume that there exists some  ∈ [] such that   = 0 for any  ∈ [] −1 .Then by (7), we have which leads to a contraction.Proposition 23.Let A be a nonnegative strongly primitive tensor and  = (A).Then, for any integer  >  > 0, we have A  > 0.
Proof.Firstly, the sufficiency is obvious.Now we show the necessity.Let  = (A).Then A  > 0 by A is strongly primitive.Let  be a positive integer such that  ≥ ; then A  > 0 by Proposition 23.Thus (A  )  = A  > 0, which implies A  is strongly primitive.

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  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.
Theorem 26.Let A be a nonnegative tensor with order  and dimension  = 2. Then A is primitive if and only if (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: ( ) .
Let A be primitive.Then (A) ≤ 2 by Theorem 10 and (A 2 ) > 0 by Proposition 8. Now we assume that (A) is not primitive; we will show A is also not primitive.
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  ≥ 3 or not.The following Example 27 shows that the necessity of Theorem 26 is false with  ≥ 3.
Example 27.Let A = (  1  2 ...  ) be a nonnegative tensor of order  and dimension  ≥ 3, where Then A is (strongly) primitive, but (A) is not primitive.
Proof.By direct calculation and Definition 2, we know that A 2 is the tensor of order ( − 1) 2 + 1 and dimension , and for any 1 ≤  ≤ , we have Obviously, A 2 is positive; then A is strongly primitive with (A) = 2 and thus A is primitive with (A) = 2.
On the other hand, by the definition of A, we have ) .
Since the associated digraph of (A) is not strongly connected, thus (A) is not primitive.
Next, we will study the strongly primitive degree of order  and dimension 2 tensors.Firstly, we discuss an example with order  = 5 and dimension  = 2 tensor as follows.
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.