Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors

Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. Furthermore, we prove that the Hadamard product of two positive semi-definite (positive definite respectively) symmetric Cauchy tensors is a positive semi-definite (positive definite respectively) tensor, which can be generalized to the Hadamard product of finitely many positive semi-definite (positive definite respectively) symmetric Cauchy tensors. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.


Introduction
Let R n be the n dimensional real Euclidean space and the set consisting of all natural numbers be denoted by N . Suppose m and n are positive natural numbers and denote [n] = {1, 2, · · · , n}. A Cauchy matrix (maybe not square) is an m × n structure matrix assigned to m + n parameters x 1 , x 2 , · · · , x m , y 1 , · · · , y n as follows: [11] C = 1 (1.1) The Cauchy matrix has been studied and applied in algorithm designing [7,10,8]. When it is a real symmetric Cauchy matrix. Stimulated by the notion of symmetric Cauchy matrices, we give the following definition.
Definition 1.1 Let vector c = (c 1 , c 2 , · · · , c n ) ∈ R n . Suppose that a real tensor C = (c i1i2···im ) is defined by Then, we say that C is an order m dimension n symmetric Cauchy tensor and the vector c ∈ R n is called the generating vector of C.
We should point out that, in Definition 1.1, for any m elements c i1 , c i2 , · · · , c im in generating vector c, it satisfies c i1 + c i2 + · · · + c im = 0, which implies that c i = 0, i ∈ [n].
By Definition 1.1, a dimension n × n real symmetric Cauchy matrix is an order 2 dimension n real symmetric Cauchy tensor. It is easy to check that every principal subtensors of a symmetric Cauchy tensor is a symmetric Cauchy tensor with a generating vector being a subvector of the generating vector of the original symmetric Cauchy tensor. In this paper, we always consider m-th order n dimensional real symmetric Cauchy tensors. Hence, it can be called Cauchy tensors for simplicity.
These papers not only established results on spectral theory and positive semi-definiteness property of structured tensors, but also gave some important applications of structured tensors in stochastic process and data fitting [3,5]. Actually, Cauchy tensors have close relationships with Hankel tensors and Hilbert tensors.
Suppose Cauchy tensor C and its generating vector c are defined as in Definition 1.1. If then Cauchy tensor C is a Hankel tensor in the sense of [14]. In general, a symmetric Cauchy tensor is not a Hankel tensor. If entries of c are defined such that then Cauchy tensor C is a Hilbert tensor according to [18].
In this paper, we are interested in the positive semi-definiteness conditions and positive definiteness conditions for even order Cauchy tensors, in addition, several spectral properties of positive semi-definite we show that the Hadamard product of two positive semi-definite (positive definite respectively) Cauchy tensors is a positive semi-definite (positive definite respectively) tensor, which can be generalized to the Hadamard product of finitely many positive semi-definite (positive definite respectively) Cauchy tensors.
In Section 3, several spectral inequalities are presented on the largest H-eigenvalue and the smallest H-eigenvalue of a Cauchy tensor. When a Cauchy tensor is positive semi-definite, bounds of its largest Heigenvalue are given. Then, for an odd order Cauchy tensor, we prove that the corresponding Z-eigenvector is nonnegative if the Z-eigenvalue is positive and the Z-eigenvector is non-positive if the Z-eigenvalue is negative. Furthermore, if the generating vector of an odd order Cauchy tensor is positive, the Cauchy tensor has nonzero Z-eigenvalues. We conclude this paper with some final remarks in Section 4. Some questions are also presented, which can be considered in the future.
By the end of the introduction, we add some comments on the notation that will be used in the sequel.

Positive Semi-definite Cauchy Tensors
An order m dimension n tensor A = (a i1i2···im ) is called positive semi-definite if for any vector x ∈ R n , it satisfies A is called positive definite if Ax m > 0 for all nonzero vector x ∈ R n . Similarly, tensor A is negative semi-definite (negative definite) if Ax m ≤ 0 (Ax m < 0 for all nonzero vector x ∈ R n ). It is obvious that minus positive semi-definite tensors (minus positive definite tensors respectively) are negative semi-definite tensors(negative definite tensors respectively). Clearly, there is no odd order nonzero positive semi-definite tensors.
In this section, we will give some sufficient and necessary conditions for even order Cauchy tensors to be positive semi-definite or positive definite. Some conditions are extended naturally from the Cauchy matrix case.
Theorem 2.1 Assume a Cauchy tensor C is of even order. Let c ∈ R n be the generating vector of C.
Then Cauchy tensor C is positive semi-definite if and only if c > 0.
Proof. For necessity, suppose that an even order Cauchy tensor C is positive semi-definite. It is easy to check that all composites of generating vector c are positive since where e i is the i-th coordinate vector of R n . So, c i > 0 for all i ∈ [n], which means c > 0.
On the other hand, assume that c > 0. For any x ∈ R n , it holds that Here the last inequality follows that m is even. By the arbitrariness of x, we know that Cauchy tensor C is positive semi-definite and the desired result holds. From the results about H-eigenvalues and Z-eigenvalues in [12], we have the following result.
Corollary 2.3 Assume that even order Cauchy tensor C and its generating vector c ∈ R n be defined as in Suppose c 1 , c 2 , · · · , c n are positive and mutually distinct, then Cauchy tensor C is positive definite.
Proof. For the sake of simplicity, without loss of generality, assume that since c > 0 and c 1 , c 2 , · · · , c n are mutually distinct. From Theorem 2.1, we know that Cauchy tensor C is positive semi-definite.
We prove by contradiction that Cauchy tensor C is positive definite when the conditions of this theorem hold. Assume there exists a nonzero vector x ∈ R n such that By the proof of Theorem 2.1, one has Thus By continuity and the fact that c 1 , c 2 , · · · , c n are mutually distinct, it holds that Repeat the process above. We obtain which is a contradiction with x = 0. So, for all nonzero vectors x ∈ R n , it holds Cx m > 0 and C is positive definite.
From this theorem, we easily have the following corollary, which was first proved in [18].

Corollary 2.4 An even order Hilbert tensor is positive definite.
From Theorem A of [6], we know that a symmetric Cauchy matrix is positive definite if and only if all the c i 's are positive and mutually distinct. In fact, the theorem below shows that conditions in Theorem 2.2 is also a sufficient and necessary condition, which is a natural extension of Theorem A of [6], by Fielder. Proof. By Theorem 2.2, we only need to prove the "only if" part of this theorem. Suppose that Cauchy tensor C is positive definite. Firstly, by Theorem 2.1, we know that We now prove by contradiction that c i 's are mutually distinct. Suppose that two elements of c are equal.
Without loss of generality, assume c 1 = c 2 = a > 0. Let x ∈ R n be a vector with elements x 1 = 1, x 2 = −1 and x i = 0 for the others. Then, one has where we get a contradiction with the assumption that Cauchy tensor C is positive definite. Thus, elements of generating vector c are mutually distinct and the desired result follows.
We denote the homogeneous polynomial Cx m as we say that f (x) is strictly monotone increasing (strict monotone decreasing respectively) in X.
The following conclusion means that the positive semi-definite property of a Cauchy tensor is equivalent to the monotonicity of a homogeneous polynomial respected to the Cauchy tensor in R n + . Proof. For sufficiency, let x = e i , y = 0 and x ≥ y. Then we have which implies that c i > 0 for i ∈ [n]. By Theorem 2.1, it holds that Cauchy tensor C is positive semidefinite.
For necessary conditions, suppose x, y ∈ R n + and x ≥ y. Then, we know that Here, the last inequality follows that x ≥ y and the fact that c i > 0, for i ∈ [n], which means that f (x) is monotone increasing in R n + and the desired result holds.
where scalars c i , i ∈ [n] are entries of generating vector c. For any x, y ∈ R n + satisfying that x ≥ y and x = y, there exists index i ∈ [n] such that Then, it holds that which implies that the homogeneous polynomial f (x) is strictly monotone increasing in R n + . Now, we give an example to show that the strictly monotone increasing property for the polynomial f (x) is only a necessary condition for the positive definiteness property of Cauchy tensor C but not a sufficient condition. Then, and the homogeneous polynomial i1,i2,i3,i4∈ [3] x i1 x i2 x i3 x i4 .
By direct computation, we know that f (x) is strictly monotone increasing in R 3 + . From Theorem 2.3, Cauchy tensor C is not positive definite.
Let r i denote the sum of the i-th row elements of Cauchy tensor C, which can be written such that where c = (c 1 , · · · , c n ) is the generating vector of Cauchy tensor C. Suppose If Cauchy tensor C is positive semi-definite, by Theorem 2.1, it is easy to check that where a = min 1≤i≤n c i ,ā = max 1≤i≤n c i . Now, before giving the next conclusion, we give the definition of eigenvalues of tensors and the definition of irreducible tensors, which will be used in the sequel.
The definition of eigenvalue-eigenvector pairs of real symmetric tensors comes from [12].
Definition 2.1 Let C be the complex field. A pair (λ, x) ∈ C × C n \ {0} is called an eigenvalue-eigenvector pair of a real symmetric tensor T with order m dimension n, if they satisfy In Definition 2.1, if λ ∈ R and the corresponding eigenvector x ∈ R n , we call λ, x H-eigenvalue and H-eigenvector respectively. Let ρ(C) denote the spectral radius of Cauchy tensor C. Then ρ(C) > 0 and The following definition is consistent with [1] and [13] respectively.

Definition 2.2
For a tensor T with order m and dimension n. We say that T is reducible if there is a nonempty proper index subset I ⊂ [n] such that Otherwise we say that T is irreducible. On the other side, by (2.1) of Definition 2.1, we have So we can takeī = l and R = rī holds. Similarly, by (2.1) and (2.2), one has which means that x i = x s and we can take i = s. Thus r = r i and the desired results follows. Theorem 2.6 Suppose even order Cauchy tensor C has positive generating vector c ∈ R n . Then C is positive definite if and only if r 1 , r 2 , · · · , r n are mutually distinct.
Proof. By conditions, all elements of c are positive, so it is obvious that r 1 , r 2 , · · · , r n are mutually distinct if and only if c 1 , c 2 , · · · , c n are mutually distinct. By Theorem 2.3, the desired conclusion follows.
Suppose A = (a i1i2···im ) and B = (b i1i2···im ) are two order m dimension n tensors, the Hadamard product of A and B is defined as which is still an order m dimension n tensor. Theorem 2.7 Let C and C ′ be two order m dimension n Cauchy tensors with generating vectors c = (c 1 , c 2 , · · · , c n ) and c ′ = (c ′ 1 , c ′ 2 , · · · , c ′ n ) respectively. Suppose m is even. If Cauchy tensors C and C ′ are positive semi-definite, then C • C ′ is a positive semi-definite tensor.
Proof. By conditions, from Theorem 2.1, it follows that For any vector x ∈ R n , by the definition of Hadamard product of tensors, we have From the arbitrariness of vector x, we know that C • C ′ is positive semi-definite.
Assume m is even. Then, C 1 • C 2 • · · · • C l is a positive semi-definite tensor.
By Theorems 2.2 and 2.3, we have the following conclusion. We omit its proof since it is similar to the proof of Theorem 2.2.
Theorem 2.8 Let C, C ′ , c and c ′ be defined as in Theorem 2.7. If Cauchy tensors C and C ′ are positive definite, then C • C ′ is a positive definite tensor.

Inequalities for Cauchy Tensors
In this section, we give several inequalities about the largest and the smallest H-eigenvalues of Cauchy tensors. The bounds for the largest H-eigenvalues are given for positive semi-definite Cauchy tensors.
Moreover, properties of Z-eigenvalues and Z-eigenvectors of odd order Cauchy tensors are also shown.
It should be noted that a real symmetric tensor always has Z-eigenvalues and an even order real symmetric tensor always has H-eigenvalues [12]. We denote the largest and smallest H-eigenvalues of Cauchy tensor C by λ max and λ min respectively. When C is a positive semi-definite Cauchy tensor, then by the Perron-Frobenius theory of nonnegative tensors [1], we have Lemma 3.1 Assume C is a Cauchy tensor with generating vector c. If the entries of c = (c 1 , c 2 , · · · , c n ) have different signs, then, Proof. From Theorem 5 of [12], we have Combining this with the fact that we have the conclusion of the lemma.
Let r, R,ā and a be defined as in Section 2. We have the following result.
Theorem 3.1 Assume even order Cauchy tensor C has generating vector c = (c 1 , c 2 , · · · , c n ). Suppose c > 0 and at least two elements of c are different. Then Proof. Suppose x ∈ R n is the eigenvector of C corresponding to λ max . By conditions, Cauchy tensor C is an irreducible nonnegative tensor and it follows x > 0 from Theorem 1.4 of [1]. Without loss of generality, let x = (x 1 , x 2 , · · · , x n ) and suppose 0 < x i ≤ 1, i ∈ [n] such that x j = 1. On the other side, by the definition of eigenvalues, from (3.1), one has and Thus, by (3.2) and (3.3), we have which can be written as Combining this with (3.1), we obtain , from which we get the desired inequalities.
In [12], Qi called a real number λ and a real vector x ∈ R n a Z-eigenvalue of tensor A and a Z-eigenvector of A corresponding to λ, it they are solutions of the following system: (3.4) Next, we will give several spectral properties for odd order Cauchy tensors.
Proof. By the condition c > 0, we know that all entries of Cauchy tensor C are positive. By definitions of Z-eigenvalue and Z-eigenvector, for any i ∈ [n], we have that Since m is odd, by (3.5), one has which implies that x ≥ 0 when λ > 0 and x ≤ o when λ < 0. Proof. By conditions, since entries of generating vector c are mutually distinct, without loss of generality, suppose 0 < c 1 < c 2 < · · · < c n .
By the continuity property, we have x 2 = 0. Repeating the process above, we get x 1 = x 2 = · · · = x n = 0, which is contradicting with the fact that x is a Z-eigenvector corresponding to λ = 0. The desired conclusion follows.

Final Remarks
In this paper, we give several necessary and sufficient conditions for an even order Cauchy tensor to be pos-