NONSINGULAR H -TENSOR AND ITS CRITERIA

. H -tensor is a new developed concept in tensor analysis and it is an extension of H -matrix and M -tensor. Based on the spectral theory of non- negative tensors, several equivalent conditions of nonsingular H -tensors are established in the literature. However, these conditions can not be used as a criteria to identify nonsingular H -tensors as they are hard to verify. In this paper, based on the diagonal product dominance and S diagonal product dom- inance of a tensor, we establish some new implementable criteria in identifying nonsingular H -tensors. The positive deﬁniteness of nonsingular H -tensors with positive diagonal entries is also discussed in this paper. The obtained results extend the corresponding conclusions for nonsingular H -matrices and improve the existing results for nonsingular H -tensors.

If the dimensions of tensor A in all directions are equal, i.e., n 1 = n 2 = · · · = n m , then A is called a square tensor, otherwise it is called a rectangular tensor. For square tensor A, if all the entries A i1i2···im are invariant under any permutation of their indices {i 1 i 2 · · · i m }, then tensor A is called symmetric [15].
Generally, tensor is a higher-order extension of matrix, and hence many concepts and related properties for matrices such as determinant, eigenvalue and singular We use I to denote m-th order n-dimensional identity tensor with entries and define the following m-order Kronecker delta The remainder of this paper is organized as follows. Some preliminaries about M -tensors, H-tensors and their properties are presented in Section 2. Based on the tensors diagonal product dominance and S diagonal product dominance, some new implementable criteria in identifying nonsingular H-tensors are established in Section 3. Further properties of nonsingular H-tensors with positive diagonal entries are investigated in Section 4. Some conclusions are drawn in the last section.
2. M -tensors, H-tensors and their properties. We first present some definitions developed in tensor analysis [2,20,28] and then introduce some kinds of specially structured tensors.
For a real m-th order n-dimensional tensor A and a scalar λ ∈ C, if there exists nonzero vector x ∈ C n such that n, then λ is said to be an eigenvalue of tensor A and x an eigenvector associated with eigenvalue λ. In particular, If x is real, then λ is also real, and we say (λ, x) is an H-eigenpair of tensor A. The largest modulus of eigenvalue of tensor A is called the spectral radius of tensor A and we denote it by ρ(A). Motivated by the characteristics of nonsingular matrices, we say a square tensor is nonsingular if its all eigenvalues are nonzero.
The following conclusion from [28] shows the linearity of the eigenvalue on the linear combination a square tensor and the identity tensor.
Lemma 2.1. For any square tensor A, scalars k and b, λ ∈ R is an eigenvalue of tensor A if and only if kλ + b is an eigenvalue of tensor kA + bI. Furthermore, they have the same eigenvectors.
When m is even and A is symmetric, tensor A is called positive (semi-)definite if (Ax m ≥ 0)Ax m > 0 for any nonzero vector x ∈ R n . By virtue of H-eigenvalue, we have the following equivalent condition for positive (semi-)definiteness of a symmetric tensor (Theorem 5, [28]). The following concept plays an important role in spectral analysis of nonnegative tensors [2].
1. An m-th order n-dimensional tensor A is called reducible, if there exists a non-empty proper index subset I ⊂ N such that Otherwise, tensor A is called irreducible.
The following specially structured tensors are extended from matrices [8,36].
Tensor A is said to be a Z-tensor if it can be written as A = cI −B, where c > 0 and B is a nonnegative tensor. Furthermore, if c ≥ ρ(B), then A is said to be an M -tensor, and if c > ρ(B), then A is said to be a nonsingular M -tensor.
It is easy to see that all the off diagonal entries of a Z-tensor are non-positive. From Lemma 2.1, scalar c can be any scalar not less than max 1≤i≤n |A ii···i |. The following conclusion gives two equivalent conditions for a Z-tensor to be a nonsingular M -tensor [8,36]. (1) The real part of any eigenvalue of tensor A is positive; (2) There exists positive vector x ∈ R n such that Ax m−1 > 0.
Clearly, for any square tensor A, its comparison tensor is a Z-tensor. Thus, tensor A can be characterized via M -tensor and hence we have the following definition.
Definition 2.4. If comparison tensor M A of tensor A is an M -tensor, then tensor A is called an H-tensor, and if comparison tensor M A is a nonsingular M -tensor, then tensor A is called a nonsingular H-tensor.
From the relationship of M -tensors and H-tensors, Proposition 2.1 can be used to characterize H-tensors. However, its conditions are hard to verify and here we introduce the following popular condition in tensor analysis [28,34,36].
and tensor A is called strictly diagonally dominant if all the inequalities hold with strict inequality.
By virtue of the diagonal dominance, Theorem 3.15 in [36] provides two criteria in identifying nonsingular M -tensors. Based on the relationship of M -tensors and H-tensors, these criteria can be applied to nonsingular H-tensors and hence we obtain the following conclusions (also see Lemmas 7,8 in [18]). Theorem 2.2. If square tensor A is strictly diagonally dominant or it is irreducible and diagonally dominant with at least one strict inequality holding in (2.1), then it is a nonsingular H-tensor.
To explore criteria with weaker conditions for nonsingular H-tensor, we give the following definition [8].
Based on this definition, we have the following equivalent condition of nonsingular H-tensors [8]. To make the paper self-contained, we give its proof.

Proposition 2.2. Tensor A is a nonsingular H-tensor if and only if
Proof. Necessity. If A is a nonsingular H-tensor, then its comparison tensor M A is a nonsingular M -tensor. From Proposition 2.1, we know that there exists vector This means that tensor M A D m−1 is strictly diagonally dominant and hence tensor AD m−1 is strictly diagonally dominant. Thus, A is generalized diagonally dominant from the definition.
Sufficiency. If tensor A is generalized diagonally dominant, then there exists positive diagonal matrix D such that tensor AD m−1 is strictly diagonally dominant. Hence, In this section, we will weaken these conditions from two aspects, one is based on the diagonal product dominance of tensors motivated by the sufficient conditions for nonsingular H-matrices established in [27], and the other is based on the S diagonal product dominance of tensor inspired by criteria for nonsingular H-matrices [5,6]. First, we have the following conclusion which has some relevance with Lemma 10 in [18].
Proof. For simplicity, we assume that the comparison tensor of tensor A is itself, i.e., Hence, A is a Z-tensor and we only need to show it is a nonsingular M -tensor. From Proposition 2.1, we only need to show that the real part of any eigenvalue of tensor A is positive under the assumption. Let (λ, x) be an eigenpair of tensor A. Then Since x = 0, we can take the largest two entries in magnitude of eigenvector x, say x i0 and x j0 . Then 2), we deduce that λ = A i0i0···i0 > 0 and the conclusion follows. Now, we assume that x j0 = 0. Consider equation   Using the facts that and A i0i0···i0 > 0, A j0j0···j0 > 0, we conclude that Re(λ) > 0. Hence A is a nonsingular M -tensor from Proposition 2.1.
Certainly, a strictly diagonally dominant tensor satisfies the assumption of Theorem 3.1. However, the converse does not necessarily hold and even it is not diagonally dominant as shown from the following example. Thus, the conclusion improves the result established in [36] Consider 3-order 2-dimensional tensor A with entries A 111 = 4, Now, we turn to consider another kind of tensor diagonal product dominance. Let S be a subset of N andS = N \S. Then we define the following multiple index sets Λ = {i 2 i 3 · · · i m | i k ∈ S for any k = 2, 3, · · · , m}, Λ = {i 2 i 3 · · · i m | i k ∈S for some k = 2, 3, · · · , m}.
Based on the above sets, we split the sum r i (A) of tensor A = (A i1i2···im ) into two parts For tensor A = (A i1i2···im ) and a partition (S,S) of N , denote , i ∈S.
Then we have the following conclusion.
then A is generalized strictly diagonally dominant and hence it is a nonsingular H-tensor.
Proof. For simplicity, we assume that the comparison tensor of A is itself. From the first inequality of (3.6), one has min i∈S h i (A) > 1. Hence we may define the following positive diagonal matrix D with diagonal entries where d > 1 is such that Now, consider tensor B = AD m−1 . It is easy to see that for any i ∈ N , and Thus for i ∈ S, if rΛ i (A) > 0, then and if rΛ i (A) = 0, then from the first inequality of (3.6), ii···i .
For i ∈S, from the second inequality of (3.6), one has This means that tensor AD m−1 is strictly diagonally dominant, and A is a nonsingular M -tensor by Proposition 2.2. If we take S = {i} and letS be its supplement set in N , then Theorem 3.2 reduces to Lemma 12 in [18]. Now, we give an analysis to the assumption in Theorem 3.2. If tensor A is strictly diagonally dominant, then for any partition of N , it holds that That is, a strictly diagonally dominant tensor satisfies the condition of Theorem 3.2. This means that Theorem 3.2 improves Theorem 2.2.
If the concerned tensor is irreducible, then the assumption in the theorem can be relaxed. To proceed, we need the following convention for x, y ∈ R, and there exists index i 0 ∈ S such that then A is a nonsingular H-tensor.
Proof. Similar to the proof of Theorem 3.2, we assume that the comparison tensor of A is itself. Then from the first inequality of (3.7) and the convention 0 0 , one has min i∈S h i (A) ≥ 1. Further, from (3.8) and (3.9), there exists d ≥ 1 such that Define positive diagonal matrix D with diagonal entries and B = AD m−1 . Then B remains irreducible as D is positively diagonal. Just as in the proof of Theorem 3.2, for any i ∈ N , it holds that For i ∈ S, if rΛ i (A) > 0, then and if rΛ i (A) = 0, then from the first inequality of (3.7), ii···i .

YIJU WANG, GUANGLU ZHOU AND LOUIS CACCETTA
Thus r Λ i (A) = 0 and hence Thus, AD m−1 is diagonally dominant with at least one strict inequality. Taking the irreducibility of tensor AD m−1 into consideration, we know that AD m−1 is generalized diagonally dominant by Proposition 2.3. Recalling Proposition 2.2, we know that A is nonsingular H-tensor.
It can easily be verified that an irreducible tensor which is diagonally dominant with one strict inequality holding in (2.1) satisfies the condition of the theorem. This means that the assumption in Theorem 3.3 is weaker than the condition of the irreducibility and the diagonal dominance with one strict inequality holding in (2.1) of a tensor. Thus, the conclusion improves the second part of Theorem 2.2.
4. Principal subtensor and nonsingular H-tensor with positive diagonal entries. In this section, we first explore the heredity of the principal subtensor of nonsingular H-tensors and then investigate the positive definiteness of nonsingular H-tensor with positive diagonal entries.
For nonnegative tensors A and B, it holds that ρ(A) ≥ ρ(B) provided that A > B ≥ 0 [34]. This can be strengthened as follows. Furthermore, the spectral radius of any principal subtensor of a nonnegative tensor is not larger than that of this tensor.
Proof. For any nonzero vector x ≥ 0, from A ≥ B ≥ 0, we know that Thus, from Theorem 5.3 in [34], one has The first conclusion follows. Let A J be a principal subtensor of tensor A whose entries are indexed by subset J of N , i.e., A J = (A i1i2···im ), i j ∈ J, j = 1, 2, · · · , m.
Certainly, Ax m−1 ≥ Bx m−1 for any nonnegative vector x ∈ R n and a similar argument to the proof of the first assertion yields that ρ(A) ≥ ρ(B).
On the other way, for principal subtensor A J , its any eigenvalue with associated eigenvector x J is also an eigenvalue of tensor B with associated eigenvector x J 0 , and vise versa. Hence, ρ(A J ) = ρ(B) from the definition of spectral radius and the second conclusion follows. From Lemma 4.1, we can readily obtain the following conclusions.
Theorem 4.1. For any nonsingular H-tensor, its any principal subtensor is also a nonsingular H-tensor.
Theorem 4.2. Let A and B be m-th order n-dimensional tensors such that Then A is a nonsingular H-tensor if B is a nonsingular H-tensor.
Now, we consider nonsingular H-tensors with positive diagonal entries. To proceed, we need the following conclusion [34]. The following conclusion shows that nonsingular H-tensors with positive diagonal entries have similar properties to nonsingular M -tensors. Then from the second inequality in (4.1), one has c > ρ(C).
On the other hand, from Lemma 2.1, λ is an eigenvalue of tensor A if and only if (c − λ) is an eigenvalue of tensor cI − A, i.e., tensor C. From ρ(C) ≤ ρ(B) < c, we know that |c − λ| < c, and hence Re(λ) > 0.
The following example shows that the converse of the theorem does not hold, i.e., for tensor A with positive diagonal entries, if the real part of its any eigenvalue is positive, then A is not necessarily a nonsingular H-tensor. This shows the difference between H-tensor with positive diagonal entries and M -tensor.
Consider 3-order 2-dimensional tensor A with and all other entries are zeros. From the definition of eigenvalue, its any eigenpair (λ, x) satisfies that

YIJU WANG, GUANGLU ZHOU AND LOUIS CACCETTA
A straightforward computing gives the spectral of tensor A, i.e., its all eigenvalues: λ A = 1, 1+2i, 1−2i. Clearly, the real part of each eigenvalue of tensor A is positive. The eigenvalues of the comparison tensor of tensor A can similarly be computed: λ M A = −1, 1, 3. From (1) of Proposition 2.1, we know that A is not a nonsingular H-tensor. From Theorem 4.3, we conclude that all H-eigenvalues of a nonsingular H-tensor with positive diagonal entries are positive. Recalling Theorem 2.1, we have the following conclusion for even order symmetric tensor (see Theorem 8 in [18]).  However, for x = (1; −1) ∈ R 2 , This means that tensor A is not positive definite. By Theorem 4.4, we know that it is not a nonsingular H-tenor.

5.
Conclusion. In this paper, we considered the nonsingular H-tensors by establishing its equivalence with generalized diagonal dominance, and investigated nonsingular H-tensors with positive diagonal entries based on the real part of tensor eigenvalue and tensor semi-positiveness. We also established some new implementable criteria in identifying nonsingular H-tensors based on the strict diagonal product dominance and S diagonal product dominance. The obtained results improve the existing results and extend the corresponding conclusions for matrices.