NEW BOUNDS FOR EIGENVALUES OF STRICTLY DIAGONALLY DOMINANT TENSORS

. In this paper, we prove that the minimum eigenvalue of a strictly diagonally dominant Z-tensor with positive diagonal entries lies between the smallest and the largest row sums. The novelty comes from the upper bound. Moreover, we show that a similar upper bound does not hold for the minimum eigenvalue of a strictly diagonally dominant tensor with positive diagonal entries but with arbitrary oﬀ-diagonal entries. Furthermore, other new bounds for the minimum eigenvalue of nonsingular M-tensors are obtained.


1.
Introduction. It is well known that the concepts of eigenvalues and eigenvectors of higher-order tensors were introduced by Qi [11] in 2005. Since then, the eigenvalue problems of high-order tensors have attracted attention of many mathematicians from different disciplines including applied mathematics branch and numerical multilinear algebra. Moreover, they also have a wide range of practical applications, see [11,9,12,13]. Applications of eigenvalues of tensors include higher-order Markov chains [10], medical resonance imaging [13,1], best-rank one approximation in data analysis [7], and positive definiteness of even-order multivariate forms in automatical control [14], etc.
Very recently, many contributions have been made on the bounds of the spectral radius of nonnegative tensor in [11,10,2,8]. Furthermore, bounds for the Z-spectral radius were given in [5] for the H-tensors. Also, in [6], He and Huang obtained the upper and lower bounds for the minimum H-eigenvalue of nonsingular M-tensors. It is important and meaningful to study the bounds for eigenvalues of a tensor. In this paper, we obtain the new bounds for the minimum eigenvalues of strictly diagonally dominant Z-tensors and nonsingular M-tensors.
The rest of the paper is organized as follows. First, we recall some definitions and theorems concerning our results. Then, the upper and lower bounds for the minimum eigenvalue of strictly diagonally dominant Z-tensors with positive diagonal entries are obtained. Moreover, we show that a similar upper bound does not hold for the minimum eigenvalue of strictly diagonally dominant tensors with positive diagonal entries but with arbitrary off-diagonal entries. Furthermore, we obtain the new bounds for the minimum eigenvalue of nonsingular M-tensors.
Denote the set of all nonnegative vectors in R n by R n + and the set of all positive vectors in R n by R n ++ . Now we recall some fundamental notions and properties on tensors. A real mth-order n-dimensional square tensor A with n m entries can be defined as follows: where i j = 1, 2, . . . , n for j = 1, 2, . . . , m.
Furthermore, a real mth-order n-dimensional tensor A is called nonnegative if all the entries a i1i2...im are nonnegative. The tensor A is called symmetric if its entries a i1i2...im are invariant under any permutation of their indices {i 1 , i 2 , . . . , i m }. The mth-order n-dimensional identity tensor, denoted by I = (δ i1i2...im ), is the tensor with entries Let A = (a i1i2...im ) be a real mth-order n-dimensional tensor, and x ∈ C n . Then Ax m−1 is a vector in C n , with its ith component defined by Let r be a positive integer. Then x [r] = (x r 1 , x r 2 , . . . , x r n ) T is a vector in C n . If there are a complex number λ and a nonzero complex vector x that are solutions of the following homogeneous polynomial equations: then we call λ an eigenvalue of A and x its corresponding eigenvector. In particular, if x is real, then λ is also real. In this case, we say that λ is an H-eigenvalue of A and x is its corresponding H-eigenvector. If x ∈ R n + , then λ is called an H + -eigenvalue of A. If x ∈ R n ++ , then λ is called an H ++ -eigenvalue of A. Moreover, we define σ(A) as the set of all the eigenvalues of A. When m is even and A is symmetric, the definition of eigenvalues of tensors was introduced by Qi [11]. When m is odd, Lim [9] used (x m−1 1 sgn(x 1 ), . . . , x m−1 n sgn(x n )) on the right-hand side instead, and the notion has been generalized by Chang, Pearson, and Zhang [3], where sgn(x) is defined by We now summarizes the Perron-Frobenius theorem for nonnegative tensors given in [3,16,15].
Let ρ(A) = max And the irreducibility of a tensor is defined as follows.
We now introduce some existing results on tensors. The following lemma was given by Qi [11]. Qi [11] introduced the Gerschogrin theorem for real symmetric tensors. Although this conclusion was proved in the case that A is a real symmetric tensor, it can be easily extended to a generic tensor of order m and dimension n. Theorem 1.4. Let A be an mth-order n-dimensional tensor. The eigenvalues of A lie in the union of n disks in C. These n disks have the diagonal elements of A as thier centers, and the sums of the absolute values of the off-diagonal elements as their radii.
By [7], we now introduce a new tensor B = (b i1i2...im ). Let A be an mth-order n-dimensional tensor and D = diag(d 1 , d 2 , . . . , d n ) be a positive diagonal matrix. Then Then we have the following lemma given in [16]. Lemma 1.5. If λ is an eigenvalue of A with corresponding eigenvector x, then λ is also an eigenvalue of B with corresponding eigenvector D −1 x; if τ is an eigenvalue of B with corresponding eigenvector y, then τ is also an eigenvalue of A with corresponding eigenvector Dy.
Zhang et al [17,4,18] extended the notion of M-matrices to higher-order tensors. Then they introduced the definitions of M-tensors and nonsingular M-tensors. Furthermore, they obtained some characterization theorems and basic properties for M-tensors and nonsingular M-tensors.
The mth-order n-dimensional tensor A is called a Z-tensor if all the off-diagonal entries are nonpositive. (1)

YINING GU AND WEI WU
A is strictly diagonally dominant if the strict inequality holds in (1) for all i.   That is, λ min (A) is a positive H + -eigenvalue of A. The first inequality of (2) is trivial and the second one is a consequence of the Gerschgorin theorem for tensors.
We now prove the last inequality of (2). Assume that A is irreducible. By Lemma 1.9, A is a nonsingular M-tensor. Hence, there exist a nonnegative tensor B and a positive real number c > ρ(B) such that Since A is irreducible, B is also irreducible. By Theorem 1.1, ρ(B) is the unique H ++ -eigenvalue of B with positive eigenvector x. By Lemma 1.3, x is also an eigenvector of A associated with λ min (A) = c − ρ(B). Let us take x = (x 1 , x 2 , . . . , x n ) T such that x i = min The last inequality of (2) holds. Now assume that A is reducible. For any ε > 0, we construct an mth-order n-dimensional tensor A ε such that A ε = A − εC, where C is an mth-order ndimensional tensor with all of its elements being 1. Then we have Hence, for any 0 < ε < (R min (A ε )/n m−1 ), A ε is a strictly diagonally dominant Z-tensor with positive diagonal entries and A ε is irreducible. Then A ε has a positive H + -eigenvalue λ min (A ε ) for any 0 < ε < (R min (A ε )/n m−1 ). Moreover, by the above conclusion, we have where R max (A ε ) = R max (A) − n m−1 ε.
Observe that A ε → A and R max (A ε ) → R max (A) as ε → 0. By the continuity of the eigenvalues as functions of the elements of the tensor, we have λ min (A ε ) → λ min (A). Hence, by considering (4) and letting ε → 0, we have λ min (A) ≤ R max (A).
Thus, we complete the proof.
The sharpness of the bounds of Theorem 2.1 depends on the dispersion of R i (A), i = 1, 2, . . . , n. If R max (A) and R min (A) are very close, then we even have an accurate approach for λ min (A).
Looking at the proof of Theorem 2.1, we can conclude that the lower bound of (2) holds for any strictly diagonally dominant tensor with positive diagonal entries but with arbitrary off-diagonal entries. By replacing R min (A) by R min (A) and λ min (A) by the minimal value of the real part of all eigenvalues of A, we have However, the upper bound of (2) cannot be extended for strictly diagonally dominant tensors whose off-diagonal entries have arbitrary sign. This fact can be illustrated by the following example.
Let A be a strictly diagonal dominant tensor of order 3 and dimension 2 with the entries defined as follows: a 111 = 2, a 122 = 1, a 222 = 3, a 211 = −2, and a i1i2i3 = 0, otherwise.
Then the eigenvalue problem of tensor A becomes: Obviously, R max (A) = 1 and from (6), we have min λ∈σ(A) Reλ = 5 2 (> R max (A) = 1). By the bounds (2) and (5), we have the following conclusion. Proof. Since R max (A) and R min (A) are the largest and the smallest row sums of M(A) respectively, by Theorem 2.1, we have λ min (M(A)) = R i (A). Now, the result follows from (5).
Thus, we complete the proof.
According to Theorem 2.1, we obtain the new bounds for the minimum eigenvalue of nonsingular M-tensors.
Theorem 2.2. Let A be a nonsingular M-tensor. Then A has a positive H +eigenvalue λ min (A) and satisfies where D = diag(d 1 , d 2 , . . . , d n ) is a positive diagonal matrix.
Proof. By the proof of Theorem 2.1, A has a positive H + -eigenvalue λ min (A). We now prove (7). Since A is a nonsingular M-tensor, by Lemma 1. Thus, we complete the proof.
The sharpness of the bounds of Theorem 2.2 depends on the distance between R max (AD m−1 ) and R min (AD m−1 ), as well as on the distance between max