A SURVEY ON THE SPECTRAL THEORY OF NONNEGATIVE TENSORS

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1 A SURVEY ON THE SPECTRAL THEORY OF NONNEGATIVE TENSORS K.C. CHANG, LIQUN QI, AND TAN ZHANG Abstract. This is a survey paper on the recent development of the spectral theory of nonnegative tensors and its applications. After a brief review of the basic definitions on tensors, the H eigenvalue problem and the Z eigenvalue problem for tensors are studied separately. To the H eigenvalue problem for nonnegative tensors, the whole Perron-Frobenius theory for nonnegative matrices is completely extended, while to the Z eigenvalue problem there are many distinctions, and are studied carefully in details. Numerical methods are also discussed. Three kinds of applications are studied: higher order Markov chains, spectral theory of hypergraphs and the quantum entanglement. 1. Introduction In 2005, independently, Lim [51] and Qi [62] introduced eigenvalues for higher order tensors. Since then, spectral theory of tensors developed rapidly, find applications or links with automatical control [62], spectral hypergraph theory [5, 6, 20, 23, 33, 35, 36, 37, 38, 61, 66], higher order Markov chains [16, 34, 48], magnetic resonance imaging [68, 70], algebraic geometry [8, 46], Finsley geometry [1], quantum entanglement [39, 65], image authenticity verification [82], etc. However, it was discovered that for general higher order tensor eigenvalue problem, even the order is low, say three or four, the problem is NP-hard [55]. This poses a big difficulty for the development of spectral theory of tensors. On the hand, it was also found, for some special classes of tensors, the eigenvalue problem is computable, and relative theory and algorithms for matrices with such special structures can be generalized to higher-order tensors without difficulty. Notably, for nonnegative tensors, i.e., tensors with nonnegative entries, the whole Perron-Frobenius theory for nonnegative matrices can be extended. Eigenvalues of nonnegative tensors have also applications in higher order Marcov chains, spectral hypergraph theory and the quantum entanglement. Thus, this attracts researchers to study the theory, algorithms and applications of eigenvalues of nonnegative tensors. Many papers appeared in this area. This paper aims to survey the progress in this area, the theory, algorithms and applications of eigenvalues of nonnegative tensors. The remainder of this paper is distributed as follows. In Section 2, we review the general spectral theory of nonnegative tensors. We then review the H-spectral theory and the Z-spectral theory of nonnegative tensors in Sections 3 and 4 respectively. In Sections 5-7, we review three applications of eigenvalues of nonnegative tensors: higher order Markov chains, spectral hypergraph theory, and the quantum entanglement. The spectral theory for nonnegative matrices has had a profound impact on both theoretical and applicable mathematics. The centerpiece of this theory resides on 1

2 2 K.C. CHANG, LIQUN QI, AND TAN ZHANG the classical Perron-Frobenius Theorems as well as some of their important consequences. In this expository article, we intend to give a brief survey to incorporate some of the most recent developments of the various spectral theories for nonnegative tensors along these lines. There is a large volume of published and unpublished work in this field; we apologize in advance if we fail to cite the work of some of our peers as any such oversight is unintentional. 2. Spectral theory of nonnegative matrices A 2-order n-dimensional real (or complex) tensor A is the n n real (or complex) matrix A = (a ij ). It can also be viewed as a linear endomorphism on R n (or C n ), hence the eigenvalue problem for A is a linear problem. In particular, the spectral radius r(a) of A is defined to be r(a) = max{ λ λ is a real or complex eigenvalue of A}. According to Gelfand s formula, r(a) = lim n An 1 n, where denotes the operator norm. Thus, r(a) is an intrinsic property of A as it is entirely determined by A itself. An m-order n-dimensional real tensor A consists of n m entries in R: A = (a i1 i m ), a i1 i m R, 1 i 1,..., i m n. However, for m > 2, the resulting eigenvalue problem is no longer a linear problem due to the different algebraic structures of the associated eigenvalue problems induced by A as we will discuss in the subsequent sections. We shall denote the set of all m-order n-dimensional tensors by R [m,n] and the set of all nonnegative m-order n-dimensional tensors by R [m,n] + henceforth. For convenience, we recall the following fundamental theorems in the spectral theory of nonnegative matrices. One may find detailed treatment of these results in the standard textbooks [3, 72]. The Perron-Frobenius Theorem, which lays the foundation for the spectral theory of nonnegative matrices, has the following two forms. Theorem 2.1. (Perron-Frobenius Theorem - Weak Form) If A is a nonnegative square matrix, then (1) r(a), the spectral radius of A, is an eigenvalue. (2) There exists a nonnegative vector x 0 0 such that Ax 0 = r(a)x 0. Definition 2.2. A square matrix A is said to be reducible if it can be placed into block upper-triangular form by simultaneous row/column permutations. A square matrix that is not reducible is said to be irreducible. Theorem 2.3. (Perron-Frobenius Theorem - Strong Form) If A is an irreducible nonnegative square matrix, then (1) r(a) > 0 is an eigenvalue. (2) There exists a positive vector x 0 > 0, i.e. all components of x 0 are positive, such that Ax 0 = r(a)x 0. (3) (Uniqueness) If λ is an eigenvalue with a nonnegative eigenvector, then λ = r(a).

3 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 3 (4) r(a) is a simple eigenvalue of A. (5) If λ is an eigenvalue of A, then λ r(a). Concerning the distribution of eigenvalues on the spectral circle we have the following: {λ C λ = r(a)}, Theorem 2.4. Let A be an irreducible nonnegative matrix. If A has k distinct eigenvalues of modulus r(a), then the eigenvalues are r(a)e i2πj/k, where j = 0, 1,, k 1. We call the number k the cyclic index of A. The special subset of nonnegative irreducible matrices known as primitive matrices plays an important role in the calculation of the spectral radius of A. We denote R n + = {(x 1,..., x n ) R n x i 0, 1 i n}, the positive cone in R n and R n ++ its interior. We recall: Definition 2.5. An irreducible nonnegative matrix A is said to be primitive if the only nonempty subset of the boundary of R n +, which is invariant under the action of A is {0}. There are many equivalent characterizations of primitivity for matrices, one of which is the following Theorem 2.6. A is a primitive matrix if and only if A has cyclic index 1. There is also a minimax characterization of the spectral radius for irreducible nonnegative matrices due to Collatz and Wielandt. Namely, Theorem 2.7. (Collatz-Wielandt) Assume A is an irreducible nonnegative n n matrix, then min max x R n ++ {i x i>0} (Ax) i x i = r(a) = max min x R n ++ {i x i>0} (Ax) i x i, This characterization of r(a) can be numerically implemented via the power method: Let A 0 be an n n irreducible matrix and let y 0 R n ++. Define Let x r = y r 1 1 y r 1, y r = Ax r, r 1. We then have yi λ r r = max 1 i n x r i yi r and λ r = min 1 i n x r i λ 0 λ 1 r(a) λ 1 λ 0. Theorem 2.8. If A is primitive, then both the sequences (x r, λ r ) and (x r, λ r ), produced by the power method, converge to (x 0, r(a)), where x 0 is the positive eigenvector corresponding to the eigenvalue r(a).

4 4 K.C. CHANG, LIQUN QI, AND TAN ZHANG 3. The H-spectral theory for nonnegative tensors To those readers, who are not familiar with tensors, we recommend [54], in which the basic definitions and properties of tensors are introduced. In 2005, Qi [62] and Lim [51] independently introduced the notion of eigenvalue problems for tensors. Here we follow the definition given in [62, 63]: Let A = (a i1 i m ) R [m,n]. Given an n-vector x = (x 1,, x n ), real or complex, we define the n-vector: ( n ) (Ax m 1 ) i := a ii2 i m x i2 x im i 2,...,i m=1 and the n-vector x [m 1] := (x m 1 1,..., x m 1 n ). 1 i n Definition 3.1. A pair (λ, x) C (C n \ {0}) is called an eigenvalue and an eigenvector of A R [m,n] if they satisfy Ax m 1 = λx [m 1]. Furthermore, we say λ is an H-eigenvalue with the corresponding H-eigenvector x (or (λ, x) is an H-eigenpair) of A if they are both real. By definition, the eigenvalues only depends on the n homogeneous polynomials (Ax m 1 ) i, i = 1, n. Thus, without loss of generality, one may assume for any fixed i {1,, n}, the (m 1)-order n-dimensional tensor (a i,i2, i m ) is symmetric. In [62], Qi proved the following conclusions on the eigenvalues of an m-order n- dimensional symmetric tensor A: Theorem 3.2. (cf. Theorems 1 and 6 [62]) (1) A number λ C is an eigenvalue of A if and only if it is a root of the characteristic polynomial φ(λ) = det(a λi), where I = (δ i1 i m ) denotes the identity tensor, i.e., { 1 if i1 = i δ i1 i m = 2 = = i m, 0 otherwise, is the Kronnecker symbol. (2) The number of eigenvalues of A is d = n(m 1) n 1. Their product is equal to det(a), the resultant of Ax m 1 = 0. (3) The sum of all the eigenvalues of A is (m 1) n 1 tr(a), where tr(a) denotes the sum of the diagonal elements of A. (4) If m is even, then A always has H-eigenvalues. A is positive definite (positive semidefinite) if and only if all of its H-eigenvalues are positive (nonnegative). (5) The eigenvalues of A lie in the following n disks: λ a i,i,,i { a i,i2,,i m : 1 i 2,, i m n, {i 2,, i m } = {i,, i}}, for i = 1,, n. Remark. The notion of symmetric tensor is referred to Definition 4.10 below. In fact, the symmetric assumption on A in [62] in this statement is superfluous. Corresponding to the n homogenous polynomials (Ax m 1 ) 1,, (Ax m 1 ) n in n variables x = (x 1,, x n ), one defines the determinant to be the resultant of these polynomials: det(a) = Res((Ax m 1 ) 1, (Ax m 1 ) n ),

5 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 5 then the characteristic polynomial becomes φ(λ) = det(a λi). All the conclusions of the above theorem continue to hold without changing the proofs. In a more general framework, Canny [7] defined the generalized characteristic polynomial (GCP), C(λ), of a system of homogeneous polynomials f 1,..., f n in the variables x 1,..., x n to be the resultant of {f 1 λx d1 1,..., f n λx dn n }, where each f i has total homogeneous degree d i. A symmetric even m order tensor A = (a i1 i m ) is called positive definite (or positive semi-definite), if Σ n i 1,,i m=1a i1 i m x i1 x im ασ n i=1x m i, α > 0, (α 0) x R n. Definition 3.3. Let A R [m,n]. We call the set σ(a) = {λ λ is an eigenvalue of A} the spectrum of A. The spectral radius ρ(a) of A is defined to be ρ(a) = sup{ λ λ σ(a)}. According to the remark following Theorem 3.2, for any A R [m,n], σ(a) is a finite set. Hence, the supremum used in the definition of ρ(a) can be replaced by the maximum. It should be noted that, unlike the spectral radius r(a) for a square matrix A, the spectral radius ρ(a) thus defined not only depends on the tensor A but also on the definition of the eigenvalue (Definition 3.1) of A. In contrast, the existence of H-eigenpairs for a general tensor A R [m,n] is significantly more challenging, we refer the interested readers to [10, 11, 88] for more detailed discussions on that front. In [51], Lim first proposed to extend the Perron-Frobenius Theorems to nonnegative tensors in this setting. He also extended the notion of irreducibility to higher order tensors as follows. Definition 3.4. A tensor A = (a i1 i m ) R [m,n] is called reducible, if there exists a nonempty proper index subset I {1,..., n} such that a i1 i m = 0, i 1 I, i 2,..., i m / I. If A is not reducible, then we call A irreducible. Due to the nonlinear nature of the eigenvalue problem (Definition 3.1) for a higher order tensor, many standard methods used in linear algebra cannot be applied directly. Surprisingly, for the family of nonnegative tensors, the main component of its spectral theory, e.g. the generalized Perron-Frobenius Theorems and some related results, can be extended completely. In 2008, Chang, Pearson, and Zhang gave the first complete proof of the following generalized Perron-Frobenius Theorems: Theorem 3.5. (cf. Theorem 1.3 [11]) If A R [m,n] +, then there exist λ 0 0 and a nonnegative vector x 0 0 such that (H) Ax m 1 0 = λ 0 x [m 1] 0. The proof of Theorem 3.5 is based on Brouwer Fixed Point Theorem. As a consequence, it directly asserts the existence of a real eigenvalue of A R [m,n] +, bypassing the resultant theory over the complex field.

6 6 K.C. CHANG, LIQUN QI, AND TAN ZHANG Theorem 3.6. (cf. Theorem 1.4 [11]) If A R [m,n] + is irreducible, then the pair (λ 0, x 0 ) in (H) satisfies: (1) λ 0 > 0 is an eigenvalue. (2) x 0 > 0, i.e. all components of x 0 are positive. (3) If λ is an eigenvalue with nonnegative eigenvector, then λ = λ 0. Moreover, the nonnegative eigenvector is unique up to a multiplicative constant. (4) If λ is an eigenvalue of A, then λ λ 0. The irreducibility condition not only implies the nonnegative eigenvector is unique, but it must be positive as well. From Theorem 3.6, we see that if A R [m,n] + is irreducible, then ρ(a) = λ 0 is an eigenvalue. Based on Theorem 3.5, the spectral radius of a nonnegative tensor is defined in [77] (which of course coincides with the more general definition ). By the simple approximation ρ(a + ɛe) ρ(a), where E denotes the unit tensor whose entries are all equal to 1, it follows that ρ(a + ɛe) ρ(a) as ɛ 0. Since A + ɛe is a positive tensor (hence irreducible), passing to the limit, Theorem 3.6 implies that ρ(a) is an eigenvalue even if A is reducible. Namely, Proposition 3.7. (cf. Theorem 2.3 [77]) For any A R [m,n] +, the spectral radius ρ(a) is an eigenvalue of A. Analogous to Theorem 2.4, Yang and Yang [77] also proved: Theorem 3.8. (cf. Theorem 3.1 [77]) Let A R [m,n] + be irreducible. If A has k distinct eigenvalues of modulus ρ(a), then the eigenvalues are ρ(a)e i2πj/k, where j = 0, 1,, k 1. We continue to call this number k the cyclic index of A. Similar to the Collatz-Wielandt Theorem 2.7, Chang et. al. extended the minimax characterization of the spectral radius ρ(a) in [11] as follow: Theorem 3.9. (cf. Theorem 4.2 [11]) Assume A R [m,n] + is irreducible, then min max x R n ++ {i x i>0} (Ax m 1 ) i x m 1 i = ρ(a) = max min x R n ++ {i x i>0} (Ax m 1 ) i x m 1. i Inspired by Theorem 3.9, Ng, Qi, and Zhou [58] proposed the following algorithm for calculating the spectral radius: (1) Choose x 0 R n ++. Let y 0 = A(x (0) ) m 1 and set k := 0. (2) Compute x (k+1) = (y(k) ) [ 1 m 1 ] (y (k) ) [ 1 m 1 ], y (k+1) = A(x (k+1) ) m 1, (y (k+1) ) i λ k+1 = min 1 i n (x (k+1) i ), m 1 (y (k+1) ) i λ k+1 = max 1 i n (x (k+1) i ). m 1

7 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 7 The iteration stops whenever λ k = λ k, which yields the largest eigenvalue λ 0 and the associated eigenvector. If the iteration does not terminate in finite time, are the sequences {λ k, x (k) } {λ k, x (k) } convergent? In order to give a complete and satisfactory answer to this question, one needs certain condition resembles primitivity for matrices. By defining the nonlinear map T A (x) := ( Ax m 1 ) [ 1 m 1 ] on R n + associated with the tensor A, Chang et. al. [12] enabled the composition of the tensor A with itself and extended the definition of primitivity to tensors. Definition An irreducible nonnegative tensor A is said to be primitive if the only nonempty subset of the boundary of R n +, which is invariant under T A is {0}. Based on the notion of primitive tensors, they proved: Theorem (cf. Theorem 4.5 [12]) If A R [m,n] + is primitive, then its cyclic index is 1. Theorem (cf. Theorems 2.7, 2.11 [12]) Let A R [m,n] +, then the following statements are equivalent: (1) A is primitive. (2) r N such that TA r(rn +\{0}) R n ++, i.e., TA r is strongly positive. (3) r N such that TA r is strictly increasing. Theorem (cf. Theorem 5.3 [12]) If A R [m,n] + is primitive, let {λ k, λ k ; x (k) } be as defined in Theorem 3.9, then λ k, λ k ρ(a) and x (k) x, i.e. x (k) converges to the eigenvector with respect to ρ(a). Moreover, Chang et. al. proved: Theorem (cf. Theorem 5.7 [12]) Let A R [m,n] + be irreducible. Both the sequences {λ k } and {λ k }, which are defined in Theorem 3.9, converge to ρ(a) for an arbitrary initial value x 0 R n +\{0} if and only if A is primitive. From the definition, it is easily seen Corollary (cf. Corollary 3.8 [12]) Let A 0 be irreducible. Then A + αi is primitive, where I is the identity tensor and α > 0. Corollary (cf. Corollary 3.7 [12]) If A 0 is essentially positive (i.e., T A is strongly positive), then A is primitive. Combining these results, the convergence results in Zhang and Qi [84] and Yang, Yang, and Li [81] follow readily. According to [56, 81], one may modify the algorithm proposed by Ng, Qi, and Zhou to any irreducible nonnegative tensor A by adding αi to A, then ρ(a) can be obtained by subtracting α in the end. Zhang, Qi and Xu [85] proved the linear convergence of the above algorithm for calculating ρ(a). Comparing the above results for tensors to the classical Perron-Frobenius Theorems for nonnegative matrices, we note the similarities as well as the differences between the two settings: similar to nonnegative matrices, the existence of a nonnegative eigenvector with nonnegative eigenvalue for any nonnegative tensor has been established. Furthermore, under Lim s irreducibility definition, the eigenvalue

8 8 K.C. CHANG, LIQUN QI, AND TAN ZHANG is positive, unique among eigenvalues with nonnegative eigenvectors and the largest in modulus. However, unlike matrices, such λ 0 may not be geometrically simple in general. Definition (cf. Definition 3.1 [11]) Let λ be an eigenvalue of A. We say λ has real geometric multiplicity q, if the maximum number of linearly independent real eigenvectors corresponding to λ equals q. If q = 1, then λ is called real geometrically simple. The following example was given in [11]. Example Let A = (a ijk ) R [3,2] + be such that a 111 = a 222 = 1, a 122 = a 211 = ɛ for 0 < ɛ < 1, and a ijk = 0 for other (ijk). Then the eigenvalue problem becomes: { x ɛx 2 2 = λx 2 1 ɛx x 2 2 = λx 2 2. We have λ 0 = 1 + ε, with eigenvectors: u 1 = (1, 1) and u 2 = (1, 1). Hence, the real geometric multiplicity of λ 0 = 1 + ε is 2. With this in mind, we seek sufficient conditions on A R [m,n] + to ensure the geometric simplicity of ρ(a). Chang in [9] studied the generalized Perron-Frobenius theorem in Banach space, and introduced the notion of semi-strong positivity for a positively 1-homogeneous, strictly increasing, compact, continuous operator. This will lead to the geometric simplicity for this kind of operators. We now review some basic terminologies. Definition Let X be a real Banach space, which has a positive cone P with nonempty interior int(p ). We write x 0 if x P ; x > 0 if x int(p ) A mapping T : X X is said to be nonnegative if 0 x implies 0 T x; it is said to be strictly positive if 0 < x implies 0 < T x; it is said to be strongly positive if 0 x and x 0 imply 0 < T x. Definition A mapping T : X X is said to be increasing if x y implies T x T y; it is said to be strictly increasing, if further, x < y implies T x < T y; it is said to be strongly increasing if x y and x y imply T x < T y. Definition Let P be a positive cone in a real Banach space X, and T : X X be a continuous map T : P P. Assume int(p ). T is called semi-strongly positive, if x P \ int(p ), there exists x P such that x, T x > 0 = x, x. where P = P \{0}, and, is the duality between X and X. The proof of the following theorem, which offers an alternative proof of Theorem 3.6 and yields the geometric simplicity result, can be found in [9] Theorem 4.8: Theorem Let T be a semi-strongly positive, increasing, 1-homogeneous, compact, continuous mapping, and r x, T x (T ) = inf sup x x P P (x) x > 0,, x where P (x) = {x P x, x > 0}. Then λ 0 = r (T ) is the unique positive eigenvalue with nonnegative eigenvector x 0. In fact, the eigenvector x 0 int(p ).

9 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 9 If λ is a real eigenvalue of T, then λ λ 0. If further, r N, such that T r is strictly increasing, then the eigenvalue λ 0 is geometrically simple. Now we return to eigenvalues for tensors. The proof of lemma 4.6 in [9] also leads to the following Lemma A nonnegative tensor A is irreducible if and only if the operator T A is semi-strongly positive. When m is even, the tensor A R [m,n] + defines a nonlinear operator on R n : T A (x) = (Ax m 1 ) 1 m 1, which maps R n + to R n +, and is 1-homogeneous. Concerning the geometric simplicity of ρ(a), without loss of generality we may assume that A is primitive In this case, T A : R n R n, which maps R n + to R n +, is an increasing, 1-homogeneous, semi-strongly positive, compact, continuous mapping. According to Theorem 3.12, r N, such that TA r is strictly increasing. Observing that when m is even, m 1 is odd, so Theorem 3.22 now implies Ax m 1 = λx m 1 T A (x) = λ 1 m 1 x, x R n. Theorem (cf. Theorem 3.1 [79]) Let A R [m,n] +. If A is irreducible and m is even, then ρ(a) is real geometrically simple. It is evident the results of Yang-Yang [77] and Pearson [60] (for essential positive tensors) on the geometric simplicity of the largest eigenvalue can be derived from Theorem The notion of irreducibility for nonnegative n n matrices A = (a ij ) have many equivalent forms. Among which, two popular definitions are commonly adopted throughout the existing literature. One of them is given by Definition 2.2. This definition of reducibility/irreducibility of a square matrix A focuses on whether or not A has nontrivial invariant coordinate subspaces, which is a combinatorially based. The other equivalent definition of irreducibility is given by the directed graph G(A) associated to A, (i.e. V = {1, 2, n} and an directed edge (i, j) E(A) if a ij > 0 ). A is irreducible if and only if G(A) is strongly connected, i.e. for any ordered pair of nodes i and j, there exists a directed path connecting i to j. When one extends the notion of irreducibility to nonnegative higher order tensors, these two approaches may lead to different consequences. Following the first approach, we have the definition of irreducibility as given in 3.4. In 2009, following the graph theory based approach, Friedland, Gaubert, and Han [25] introduced the notion of weakly irreducible tensors. It is defined as follow. Given a nonnegative tensor A = (a i1 i m ) R [m,n] +, it is associated to a directed graph G(A) = (V, E(A)), where V = {1, 2,, n} and a directed edge (i, j) E(A) if there exists indices {i 2,, i m } such that j {i 2,, i m } and a ii2 i m > 0, i.e., (3.1) Σ j {i2,,i m}a ii2, i m > 0. Definition A nonnegative tensor A R [m,n] + is called weakly irreducible if the associate directed graph G(A) is strongly connected.

10 10 K.C. CHANG, LIQUN QI, AND TAN ZHANG It is equivalent to say (see Hu in [30]) that the matrix M(A) = (m ij ) is irreducible, where m ij = a ii2, i m. j {i 2,,i m} It is straightforward to show an irreducible nonnegative tensor is weakly irreducible, but the following example is a weakly irreducible tensor which is reducible according to Definition 3.4. Example (cf. [80]) Let A R [4,3] + be given by a 1111 = a 1123 = a 2223 = a 3113 = 1 and a ijkl = 0 elsewhere. In [25], Friedland et. al. discovered that a series of results obtained by Nussbaum [44, 45], Burbanks, Nussbaum, and Sparrow [4], and Gaubert and Gunawardena [27] etc. on order preserving mappings as well as on positively 1-homogeneous monotone functions can be applied to the nonnegative tensors setting. Applying these results, they reproved Theorem 3.6 under the weakly irreducible condition: Theorem Assume that A R [m,n] + is weakly irreducible. Then there exists a unique positive H-eigenvector with positive eigenvalue. It is imperative to distinguish the results of Theorem 3.27 and Theorem 3.6. Since the weak irreducibility assumption is weaker than irreducibility, Theorem 3.27 establishes the existence of a positive H-eigenpair under a weaker assumption. However, Theorem 3.6 asserts the positive H-eigenvector is unique up to a nonnegative multiplier in R n + \ {0}, and there are no H-eigenvectors on the boundary of the positive cone. Whereas, Theorem 3.27 asserts that the positive H-eigenvector is unique in R n ++, but does not rule out the possibility of an H-eigenvector on the boundary of R n +. From a topology point of view, there are two essentially different existence proofs for the classical Perron-Frobenius Theorem for nonnegative matrices and its generalizations to monotone operators in the literature: one approach is based on Brouwer Fixed Point Theorem. This idea can be traced back to Alexandroff and Hopf in In 1940, Rutman continued in this vein to reprove Jentsch s theorem on integral equations, which is an infinite dimensional analog of the Perron-Frobenius Theorem, by using Schauder fixed point theorem. The other approach is to reduce the eigenvalue problem to a contraction mapping by using the Hilbert projective metric. This idea can be traced back to Samuelson (1956), Birkhoff (1957), Thompson (1963), and Nussbaum(1988). The proof of Theorem 3.6 follows Brouwer fixed point theorem, and the proof of Theorem 3.27 follows the approach of the contraction mapping. Friedland, Gaubert, and Han [25] also proved the convergence of the power algorithm proposed by Ng, Qi, and Zhou continues to hold for a weakly primitive nonnegative tensor (i.e, in the definition of a primitive tensor, the irreducibility condition is replaced by weak irreducibility). While many important properties of ρ(a) are still enjoyed by the nonnegative weakly irreducible tensors, some others may be lost nevertheless under this weaker notion. In a recent paper [80], Yang and Yang showed using the Example 3.26, x 1 = ( , , ) and x 2 = ( , , ) are both approximated eigenvectors corresponding to ρ(a) So, ρ(a) of a nonnegative even order weakly irreducible is not necessarily real geometrically simple, contrasting to the conclusion of Theorem 3.24.

11 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 11 In addition to the results mentioned above, there are various extensions on the positiveness of the tensors, e.g., Zhang and Qi [84] studied the weakly positive tensors; Hu, Huang, and Qi [31] studied the strictly nonnegative tensors. Further developments can be found in [56, 67, 81, 76, 93, 85, 95, 96]. Extensions to rectangular nonnegative tensors, essentially nonnegative tensors, copositive tensors, completely positive tensors and M-tensors can be found in [14, 15, 32, 67, 69, 78, 83, 86, 87, 93]. 4. The Z-spectral theory for nonnegative tensors Parallel to the eigenvalue (resp. H-eigenvalue) problem of A R [m,n], there are other types of eigenvalue problems of A, e.g. [10, 51, 62, 63]. In particular, we will investigate the following: Definition 4.1. (cf. [63]) Let A R [m,n]. A pair (λ, x) C (C n \ {0}) is called an E-eigenvalue and E-eigenvector (or simply E-eigenpair) of A if they satisfy the equation { Ax m 1 = λx, x x = 1 We call (λ, x) a Z-eigenpair if they are both real. Independently, Lim also defined eigenvalues for tensors in [51]. Lim defined eigenvalues for general real tensors in the real field. The l 2 -eigenvalues of tensors defined by Lim are Z-eigenvalues of Qi [62], while the l m -eigenvalues of tensors defined by Lim are H-eigenvalues in Qi [62]. Notably, Lim proposed a multilinear generalization of the Perron-Frobenius theorem based upon the notion of l m -eigenvalues (H-eigenvalues) of tensors. When m is even, the E-characteristic polynomial of A is defined to be the resultant ψ A (λ) = Res x (Ax m 1 λ(x x) m 2 2 x). It is a polynomial of λ. When m is odd, it is defined in [63] to be the resultant: ψ A (λ) = Res x,x0 (Ax m 1 λx m 2 0 x, x x x 2 0). We say that A is regular if the following system has no nonzero complex solutions: { Ax m 1 = 0, x x = 0 For a regular tensor, for all m, the E-characteristic polynomial reads as ψ A (λ) = Res x,x0 (Ax m 1 λx m 2 0 x, x x x 2 0). In [62], Qi also established the following conclusions on the E-eigenvalues of an m-order n-dimensional symmetric tensor A: Theorem 4.2. (cf. Theorems 1-4 [63]) (1) When A is regular, a complex number is an E-eigenvalue of A if and only if it is a root of its E-characteristic polynomial. (2) The Z-eigenvalues always exist. An even order symmetric tensor is positive definite if and only if all of its Z-eigenvalues are positive. (3) If A and B are orthogonally similar, then they have the same E-eigenvalues and Z-eigenvalues.

12 12 K.C. CHANG, LIQUN QI, AND TAN ZHANG (4) If λ is the Z-eigenvalue of A with the largest absolute value and x is a Z- eigenvector associated with it, then λx m is the best rank-one approximation of A, i.e., A λx m F = A 2 F λ2 = min{ A αu m F : α R, u R n, u 2 = 1}, where F is the Frobenius norm. Theorem 4.2 (4) indicates that Z-eigenvalues play an important role in the best rank-one approximation. The best rank-one approximation of higher order tensors has extensive engineering and statistical applications, such as Statistical Data Analysis [22, 41, 89]. Both of the eigenvalue problem and the E eigenvalue problem for tensors are nonlinear in nature with various new and important applications in numerical multilinear algebra, image processing, higher order Markov chains, spectral hypergraph theory, and the study of quantum entanglement, etc. However, their chief difference lies in that the eigenvalue problem is equivalent to finding nontrivial solutions of a system of homogeneous multivariate polynomial equations of the same degree, whereas the E-eigenvalue problem is equivalent to finding nontrivial solutions of a system of inhomogeneous multivariate polynomial equations. It was shown in a recent paper by Li, Qi, and Zhang [46] that the E-characteristic polynomial of a given tensor A R [m,n] (not necessarily symmetric) is in fact invariant under the action of the orthogonal group. As a consequence, all E/Z-eigenvalues of A are orthogonal invariants of A. The significance of the orthogonal invariance of the E/Z-eigenvalues indicates many results on the Z-eigenvalues of nonnegative tensors will remain valid for a broader class of tensors which are not necessarily nonnegative themselves but are orthogonally similar to nonnegative tensors. Unfortunately, the H-eigenvalues are not orthogonally invariant. Ni, Qi, Wang, and Wang [43] showed: For even m > 2, the degree of the E- characteristic polynomial ψ A (λ) of a generic tensor A is d E = (m 1)n 1 m 2 = (m 1) n 1 + (m 1) n (m 1) + 1. They conjectured that the assumption on m being even is not necessary. Later, Cartwright and Sturmfels proved the conjecture in [8]. Furthermore, Cartwright and Sturmfels observed (cf. Proposition 3.4 [8]) the following implications of the relationship between the E-eigenvalues and the E-characteristic polynomial ψ A (λ) as follows: (1) The set of all E-eigenvalues of A consists of all complex numbers, which implies (2) The set of all E-eigenvalues of A is infinite, which implies (3) The E-characteristic polynomial ψ A (λ) vanishes identically. In the E-eigenvalue problem, the regular assumption for a tensor is very important, this can be seen in the following example [8]: Example 4.3. Let A = (a ijk ) C [3,2], where a 111 = a 221 = 1, a 112 = a 222 = i, a ijk = 0 otherwise.

13 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 13 We solve the system { x ix 1 x 2 = λx 1, x 1 x 2 + ix 2 2 = λx 2 It is easily seen that all λ 0 are E-eigenvalues of A. The striking difference between the eigenvalues of A and the E-eigenvalues of A is depicted by the following result due to Cartwright and Sturmfels: Theorem 4.4. (cf. Proposition 3.3 [8]) The set of E-eigenvalues of A is either a finite set or it consists of all complex numbers in the complement of a finite set. The following versions of the generalized Perron-Frobenius Theorems for the Z- eigenvalues as well as other types of eigenvalues of a nonnegative tensor A were established in Chang, Pearson, and Zhang [11, 13]. Theorem 4.5. If A R [m,n] +, then there exists a Z-eigenvalue λ 0 0 and a nonnegative Z-eigenvector x 0 0 of A such that Ax0 m 1 = λ 0 x 0. Theorem 4.6. If A R [m,n] + is irreducible, then the pair (λ 0, x 0 ) in Theorem 4.5 satisfy: (1) The eigenvalue λ 0 is positive. (2) The eigenvector x 0 is positive, i.e. all components of x 0 are positive. However, unlike the H-eigenpairs for a nonnegative irreducible tensor A, neither the positive Z-eigenvalue nor the associated positive Z-eigenvector of A has to be unique in general; see Errata [11] and Example 2.7 in [13]. Similar to the notion of spectrum set and spectral radius for a tensor A R [m,n], Chang et. al. in [13] introduced Definition 4.7. Let A R [m,n]. We define the Z-spectrum of A to be Z(A) = {λ R λ is a Z-eigenvalues of A}. Assume Z(A), ϱ(a) := sup { λ λ Z(A) } is called the Z-spectral radius of A. According to Theorem 4.5, for a nonnegative tensor A R [m,n] +, Z(A), so the Z-spectral radius ϱ(a) is well defined. But, different from the H-spectral radius ρ(a), ϱ(a) may not be itself a positive Z-eigenvalue of A; see Example 3.3 in [13]. Based on the above observations, we introduce the following Definition 4.8. (cf. [13]) Let A R [m,n] +. We define the nonnegative spectrum of A, Λ(A) = {λ 0 x R n + S n 1 satisfying Ax m 1 = λx}, where S n 1 is the standard unit sphere in R n. It is important to notice the set Λ(A) is not necessarily a finite set in general as shown by Example 3.5 [13]. Proposition 4.9. (cf. Proposition 3.6 [13]) Let A R [m,n] +. Then Λ(A) is a compact subset.

14 14 K.C. CHANG, LIQUN QI, AND TAN ZHANG For A R [m,n] +, we define λ = max{λ Λ(A)}. Besides the family of nonnegative tensors, the family of (weakly) symmetric tensors constitutes another special yet important object to study. We recall the following from [10, 62]. Definition A R [m,n] is called symmetric if a i1 i m = a σ(i1 i m) for all σ S m, where S m denotes the permutation group of m indices. It was first called the super-symmetric tensor, but was suggested changing to be symmetric tensor by [19]. In that paper, it also demonstrates equivalence to the usual coordinate-free definition of a symmetric tensor in algebra. The notion of weakly symmetric tensors is introduced in [10]. Definition A R [m,n] is called weakly symmetric if the associated homogeneous polynomial n Ax m = f A (x) := a i1i 2 i m x i1 x i2 x im satisfies f A (x) = max m 1. i 1,i 2,...,i m=1 It is also shown in [10] that a symmetric tensor is necessarily weakly symmetric, but the converse is not true in general. Furthermore, if A R [m,n] is weakly symmetric, by homogeneity, we find Ax m = f A (x) = 1 m f A(x), x = Ax m 1, x, where, denotes the standard inner product on R n. We define and we have: λ := max f A (x) = max Ax m x S n 1 x S n 1 Theorem (cf. Theorem 3.11 [13]) Assume A R [m,n] + is weakly symmetric. Then λ = λ = ϱ(a). Definition (cf. [13]) Let A R [m,n] +. We say A is non-degenerate if for all x R n + \ {0}, (Ax m 1 ) i and x i do not vanish simultaneously for all i {1,, n}. It is straightforward to show if A is irreducible, then A is non-degenerate; see section 4 [13] for detail. However, the converse is not true in general, e.g. Example 4.7 [13]. Definition (cf. [13]) Let A R [m,n] + be non-degenerate. We define the following two functions for all x R n + \ {0}: (Ax m 1 ) i ν (x) := min and ν (Ax m 1 ) i (x) := max. 1 i n x i 1 i n x i In the definition it may happen ν (x) =.

15 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 15 Definition (cf. [13]) We define ϱ := sup x R n ++ S n 1 ν (x) and ϱ := inf x R n ++ Sn 1 ν (x). We then have a similar max-min characterization of ϱ(a): Theorem (cf. Theorem 4.5 [13]) Assume A R [m,n] + is weakly symmetric and irreducible. Then (1) Λ(A) is contained in the closed interval [ϱ, ϱ ], i.e. Λ(A) [ϱ, ϱ ]. (2) ϱ(a) = λ = λ = ϱ. In order to compute ϱ(a), we adapt an iterative algorithm known as the Shifted Symmetric Higher-Order Power Method (SS-HOPM), proposed by Kolda and Mayo [42] when m is even. Although the algorithm as well as its convergence analysis are given under the assumption A R [m,n] being symmetric, the entire process nonetheless continues to work successfully when we only assume A is weakly symmetric. We refer the interested reader to [42] for a more in-depth discussion on this subject. We now adapt the SS-HOPM (cf. Algorithm 2 [42]) as follows. Given a weakly symmetric tensor A R [m,n] +. Step 0. Choose x (0) R n + \ {0}, set λ 0 = Ax m (0), and choose the shift constant n α = m a i1...i m, i 1,,i m=1 where γ is the ceiling function, i.e. it equals the smallest integer no less than γ. Set k := 0. Step 1. Set y (k+1) := Ax m 1 (k) + αx (k). Step 2. Compute x (k+1) := y (k+1) y (k+1) λ k+1 := Ax m (k+1). Since, α is large enough, the function f A (x) + α x m becomes convex on R n. For even m, the convergence of the sequence λ k is guaranteed. 5. Higher order Markov chains Eigenvalue theory for nonnegative matrices plays an important role in the study of finite Markov chain. A finite Markov chain is a stochastic process X 0, X 1,, with values in {1,..., n}. Let Prob(X t+1 = i X t = j) = p ij. We call P = (p ij ) the transition probability matrix; clearly p ij 0,and n p ij = 1 j = 1,, n. i=1 A nonnegative matrix satisfying the above conditions is called a stochastic matrix. Let ξ t R n be the distribution of X t, i.e., n (ξ t ) i = Prob(X t = i), and (ξ t ) i = 1, i=1

16 16 K.C. CHANG, LIQUN QI, AND TAN ZHANG then we have ξ t+1 = P ξ t and n (ξ t+1 ) i = 1. If ξ R n is the eigenvector of P, with ξ 0 and n i=1 ξ i = 1, then P ξ = ξ. In this sense, ξ corresponds to an invariant distribution of the Markov chain. In case P is irreducible, according to the classical Perron-Frobenius Theorem, ξ corresponds to the largest eigenvalue 1, which is simple. One can show i=1 ξ t ξ, as t. In this sense, ξ is also called the equilibrium distribution. Motivated by the demand of multi-relational data mining, M. Ng et. al. discovered intriguing new connections of the Z-eigenvalue problem to the transition probability tensors of higher order Markov chains in a series of their recent work [48, 49, 50]. They proposed a framework (HAR) that can be used to compute hub, authority, and relevance scores in multi-relational data query. The basic facts of higher order Markov chains can be found in [18]. A higher-order Markov chain is an extension of the finite Markov chain, in which the stochastic process X 0, X 1, with values in {1, 2,, n} has the transition probabilities: (5.1) 0 p i1i 2 i m = Prob(X t = i 1 X t 1 = i 2,..., X t m+1 = i m ) 1 where (5.2) n p i1,i 2,,i m = 1, 1 i 2,..., i m n. i 1=1 Thus, we have a tensor P R [m,n] + consisting of n m entries in between 0 and 1: P = (p i1i 2...i m ), 1 i 1, i 2,..., i m n, satisfying 5.2. We call it a transition probability tensor. Let the probability distribution at time t be ξ (t) n, where n n = {x = (x 1,, x n ) R n + : x j = 1}. Then we have (5.3) ξ (t+m) = Pξ (t+m 1) ξ (t) := for t = 1, 2,. Assume that ( n i 2 i m=1 j=1 ) n p i,i2 i m ξ (t+m 1) i 2 ξ (t) i m n i=1 (5.4) lim t ξ (t) = ξ. Then we have ξ n and we may call ξ the stationary probability distribution of the higher-order Markov chain. It follows that ξ n is an eigenvector of P with eigenvalue 1, i.e, (5.5) Pξ m 1 = ξ. Although the eigenvalue problem (5.5) is different from the Z-eigenvalue problem for P :

17 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 17 { Px m 1 = λx, x x = 1; they share the same eigenvector (with a positive constant multiplier), but they may correspond to different eigenvalues [16]. Consequently, if P is irreducible, then a positive solution of (5.5) exists. This is also proved by Li and Ng [48]. Moreover, they gave sufficient conditions for the uniqueness of the fixed point problem (5.5) inside the simplex n. Under the same conditions (essentially to ensure P being a contraction mapping on n ), Li and Ng [48] established linear convergence of the power method (5.6) y (k+1) = P (y (k)) m 1 Although it has been noted in the previous section that for an arbitrary irreducible nonnegative tensor the largest Z-eigenvalue may correspond to more than one positive Z-eigenvector (or equivalently, the fixed points of P inside n need not be unique for an arbitrary irreducible nonnegative tensor), it was unclear whether every irreducible transition probability tensor has a unique fixed point, until a counter example appeared in [16]: Example 5.1. Let P R [4,2] + be the positive (hence irreducible) transition probability tensor defined by p 1111 = 0.872, p 1112 = p 1211 = , p 1121 = , p 1122 = 0.616, 3, p 1212 = 0.616, p 1221 = , p 1222 = 0.072; p 2111 = 0.128, p 2112 = p 2211 = , p 2121 = , p 2122 = 2.384, 3, p 2212 = 2.384, p 2221 = , p 2222 = The fixed points of P (5.5) are solutions the following system: (Px 3 ) 1 = 0.872x x 2 1x x 1 x x 3 2 = x 1, (Px 3 ) 2 = 0.128x x 2 1x x 1 x x 3 2 = x 2, x 1 + x 2 = 1, x 1, x 2 [0, 1]. It is easy to verify that both v 1 = (0.2, 0.8), v 2 = (0.6, 0.4) satisfy the system. Consequently, finding sufficient conditions to ensure the uniqueness of the fixed point for an irreducible transition probability tensor becomes an important task to be carried out. In this direction, we mention the works of Hu, Qi [34] and Chang, Zhang [16]. Another important problem in this aspect is the convergence of the probability distribution ξ (t) as time t, i.e.(5.4). For second-order Markov chain, under some additional conditions, Hu and Qi [34] established the convergence of (5.4).

18 18 K.C. CHANG, LIQUN QI, AND TAN ZHANG 6. Applications to spectral hypergraph theory Spectral graph theory is a well developed area of graph theory with applications to combinatorics, computer science, neural networks, and social sciences, etc. A graph can be encoded by its adjacency matrix M. A main theme in spectral graph theory is to study the properties of the graph and the eigenvalues or singular values of M. Lim [51] pointed out that a potential application of the largest eigenvalue of a nonnegative tensor is on hypergraphs. However, to extend the existing results and methodology from spectral graph theory to spectral hypergraph theory, a more sophisticated approach is required; namely via tensors. We adopt the following standard definition of hypergraphs, for more details, we refer to [2, 20, 21, 33, 35, 36, 37, 38, 47, 61, 66, 74]. Definition 6.1. A hypergraph H is a pair (V, E), where E P(V ), the power set of V. The elements of V = V (H) are called vertices and the elements of E = E(H) are called edges. A hypergraph H is said to be k-uniform for an integer k 2 if, for all e E(H), the cardinal number of the subset, e = k. The term k-graph is often used in place of k-uniform hypergraph. By definition, a 2-graph is a graph. Definition 6.2. The adjacency tensor A H for a uniform m-graph H = (V, E), denoted A H = (a i1,,i m ) R [m,n], where n is the number of the set V, is the symmetric tensor given by { 1 1, if i1, i a i1,,i m = m E (m 1)! 0, otherwise Characteristic polynomial plays an important role in the spectral graph theory. In the study of the spectrum of hypergraphs, the characteristic polynomial φ(λ) = det(a λi) for the tensor A is used extensively. Since our subject is the spectral theory for nonnegative tensors, we shall focus on the applications of results obtained in previous sections. Definition 6.3. H is said to be m-regular if every v V is adjacent to exactly m hyper-edges. A hypergraph H = (V, E) is said to be k-partite or k-colorable if there exists a partition of the vertices V = V 1 V k such that for any k vertices i 1,, i k with a i1 i k 0, i 1,, i k must each lie in a distinct V i for 1 i k. Let λ max be the H eigenvalue (in the sense of Definition 3.1) of a m-graph H with largest modulus. It follows from Theorem 3.6 that Theorem 6.4. (cf. Theorem 3.7 [20]) For any non-empty m-graph H, λ max can be chosen to be a positive real number. If H is connected, then a corresponding eigenvector x can be chosen to be strictly positive. This result is subsequently generalized by Pearson and Zhang to multi-graphs using a different approach. Theorem 6.5. (cf. Theorem 1 [61]) Let H be a connected m-multigraph on n vertices, then: (1) There exists (λ 0, x 0 ) R ++ R n ++ an H-eigenpair of A H, where λ 0 = ρ(a H ) = max x S + A H x m is the H-spectral radius of A H, where S + = {x R n + n i=1 xm i = 1}.

19 SPECTRAL THEORY FOR NONNEGATIVE TENSORS 19 (2) The positive eigenvector x 0 is unique (up to a positive multiplier) in R n ++. see also [53]. Further discussions on the H as well as Z-eigenvalues/eigenvectors of the adjacency tensor can be found in [20] and [61]. Recently, there has been rapid developments in the discovery of spectral properties of the Laplacian and signless Laplacian of uniform hypergraphs. According to Qi et. al. [35], we have the following: Definition 6.6. For a k-uniform hypergraph H = (V, E) on n vertices, let D be a k-th order n-dimensional diagonal tensor with its diagonal element d i i being d i, the degree of vertex i, for all i [n]. Then L := D A H is the Laplacian tensor of the hypergraph H and Q := D + A H is the signless Laplacian tensor of the hypergraph H. Both the Laplacian tensor and the signless Laplacian tensor of the hypergraph H have exhibited promising spectral properties analogous to the Laplacian matrix for graphs. For a more in depth exposition, we refer the interested readers to [35, 36, 74]. For a hypergraph H, a function f : V (H) [r] is a (weak) proper r-coloring of H if for every edge e = {v 1, v 2,..., v k } there exist i j such that f(v i ) f(v j ). The (weak) chromatic number of H, denoted χ(h), is the minimum r such that H has a proper r-coloring. In the same paper, Cooper and Dutle showed: Theorem 6.7. (cf. Theorem 3.9 [20]) For any k-graph H, χ(h) λ max + 1. Given an undirected graph G and a positive integer k ω(g), the clique number, i.e, the maximum size of a clique (a subset of mutually adjacent vertices in V) in G, one can build a hypergraph H, which is called the k-clique (k + 1)-graph of G having k-cliques of G as vertices and (k + 1)-cliques of G as edges. By using the spectral radius of the hypergraph H, Bulo and Pelillo [5, 6] obtained new upper and lower bounds for ω(g): Theorem 6.8. (cf. Theorem 5 [5]) Let G be an undirected graph with clique number ω(g) and H be a k-clique (k + 1)-graph of G with spectral radius r(h). Then ω(g) r(h) k! Theorem 6.9. (cf. Theorem 6 [5]) Let G be an undirected graph with clique number ω(g) and H a k-clique (k + 1)-graph of G with spectral radius r(h) and Perron eigenvector x 0. Then ω(g) ψ 1 r(h) ( k! x 0 k+1 ), k where ψ k (x) = (x k)(ck n ) 1 k. + k. Although the Laplace matrix, which plays an important role in spectral graph theory, is not a nonnegative matrix, after modifications, it can be reduced to a nonnegative matrix. The Laplace matrix for an undirected graph G is symmetric and semi-positive definite, the smallest eigenvalue µ 1 (G) is 0, and the multiplicity of 0 is the number of connected components of G. Moreover, the second smallest Laplace eigenvalue µ 2 (G) is called the algebraic connectivity of the graph G, one

20 20 K.C. CHANG, LIQUN QI, AND TAN ZHANG has: µ 2 (G) > 0 if and only if G is connected. The above notions and results are extended to k-uniform hypergraphs in [33, 47], where the Z-eigenvalues for tensors are used. 7. Applications to quantum entanglement The quantum entanglement problem is a central problem in quantum information [57]. In 2003, Wei and Goldbart [73] introduced geometrical measure for quantum entanglement. Wei and Goldbart [73] conjectured that the nearest separable state for a symmetric state can be chosen to be symmetric. Hayashi, Markham, Murao, Owari, and Virmani [29] proved the conjecture for symmetric states with nonnegative amplitudes. Hübener, Kleinmannn, Wei, González-Guillén, and Gühne [40] proved the conjecture in general. Also see Orús, Dusuel, and Vidal [59]. The computation of the symmetric pure states with nonnegative amplitudes was carried out by Wei and Goldbart [73] for some ground states, and systematically for symmetric pure multipartite qubit states by Chen, Xu, and Zhu [17]. Very recently, Hu, Qi, and Zhang [39] established the link between the geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors. An m-partite pure state Ψ of a composite quantum system is an element in a Hilbert tensor product space H = m k=1 H k, with Ψ Ψ = 1, where the dimension of H k is d k for k = 1,..., m. A separable (Hartree) m-partite state Φ H has the form Φ = m k=1 φ(k) with φ (k) H k and φ (k) = 1 for k = 1,..., m. A state is called entangled if it is not separable. For a given m-partite pure state Ψ H, its nearest separable state Φ = m k=1 φ(k) can be described by the maximal overlap: (7.1) G(Ψ) = The geometric measure is defined as [73] max Φ = Ψ Φ. m k=1 φ(k) E G ( Ψ ) = 1 G(Ψ) 2. It was shown that the maximal overlap in (7.1) is equal to the largest entanglement eigenvalue λ [73, 65]: ( ) Ψ j k φ(j) = λ φ (k), ( (7.2) j k ) φ(j) Ψ = λ φ (k), φ (k) = 1, k = 1,..., m. A state Ψ H = m k=1 H k is called nonnegative ( if there exist ) orthonormal bases { e (k) i } d k i=1 for H k such that a i1...i m := Ψ e (1) i 1 e (m) i m 0 for all i j = 1,..., d j and j = 1,..., m. The d 1 d m tensor consisting of a i1...i m is denoted by A Ψ. When H 1 = = H m, A Ψ is symmetric if and only if Ψ is symmetric in the sense of quantum information. When Ψ is symmetric, (7.1) reduces to [40] (7.3) G(Ψ) = max Ψ Φ. Φ = φ m The following theorem in [39] established the link between the geometric measure of entanglement of pure states with nonnegative amplitudes and the largest Z- eigenvalues of nonnegative tensors.