Real eigenvalues of nonsymmetric tensors

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1 Comput Optim Appl (208) 70: 32 Real eigenvalues of nonsymmetric tensors Jiawang Nie Xinzhen Zhang 2 Received: 30 May 207 / Published online: 4 December 207 Springer Science+Business Media, LLC, part of Springer Nature 207 Abstract This paper discusses the computation of real Z-eigenvalues and H- eigenvalues of nonsymmetric tensors. A generic nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. The number of H-eigenvalues is finite for all tensors. We propose Lasserre type semidefinite relaxation methods for computing such eigenvalues. For every tensor that has finitely many real Z-eigenvalues, we can compute all of them; each of them can be computed by solving a finite sequence of semidefinite relaxations. For every tensor, we can compute all its real H-eigenvalues; each of them can be computed by solving a finite sequence of semidefinite relaxations. Keywords Tensor Z-eigenvalue H-eigenvalue Lasserre s hierarchy Semidefinite relaxation Mathematics Subject Classification 5A8 5A69 90C22 Introduction For positive integers m and n, n 2,...,n m, an m-order and (n, n 2,...,n m )- dimensional real tensor can be viewed as an array in the space R n n 2 n m. Such a tensor A can be indexed as B Xinzhen Zhang xzzhang@tju.edu.cn Jiawang Nie njw@math.ucsd.edu Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA 2 School of Mathematics, Tianjin University, Tianjin , China 23

2 2 J. Nie, X. Zhang A = (A i...i m ), i j n j, j =,...,m. (.) When n = = n m = n, A is called an m-order n-dimensional tensor. In such case, the tensor space R n n m is denoted as T m (R n ). A tensor in T m (R n ) is said to be symmetric if its entries are invariant under permutations of indices [i.e., A i...i m = A j... j m whenever (i,...,i m ) is a permutation of ( j,..., j m )]. The subspace of symmetric tensors in T m (R n ) is denoted as S m (R n ).ForA T m (R n ) and x := (x,...,x n ), we use the notation Ax m := A i i 2...i m x i x i2 x im, i (,...,i m n ) Ax m (.2) := A ji2...i m x i2 x im. i 2,...,i m n j=,...,n Note that Ax m is an n-dimensional vector. First, we give some definitions of tensor eigenvalues, which can be found in [2,3]. Definition. For a tensor A T m (R n ), a pair (λ, u) R R n is called a real Z-eigenpair of A if Au m = λu, u T u =. (.3) (The superscript T denotes the transpose.) Such λ is called a real Z-eigenvalue, and such u is called a real Z-eigenvector associated to λ. Definition.2 For a tensor A T m (R n ), a pair (λ, u) R R n is called a real H-eigenpair of A if Au m = λu [m ], u = 0. (.4) (The symbol u [m ] denotes the vector such that (u [m ] ) i = (u i ) m for i =,...,n). Such λ is called a real H-eigenvalue, and such u is called a real H-eigenvector associated to λ. Tensor eigenvalues are defined in Lim [2]. For real symmetric tensors, they are also defined in Qi [3]. The Z-eigenvalus and H-eigenvalues have applications in signal processing, control, and diffusion imaging [35,37,38]. For recent work on tensor computations, we refer to [22,30,34]. In particular, Z-eigenvalues have important applications in higher order Markov chains, which are shown in Sect Definitions. and.2 are about real eigenvalues. Complex eigenvalues can be similarly defined, if the complex values are allowed for (λ, u). We refer to the work [3,32]. There also exist other types of tensor eigenvalues in [4,5]. In this paper, we focus on the computation of real Z/H-eigenvalues. When a tensor is symmetric, the computation of real Z/H-eigenvalues is discussed in [5]. For the case (m, n) = (3, 2), computing the largest Z-eigenvalues is discussed in [36]. Shifted power methods are proposed in [5,40]. In [24], a method is proposed to find best rank- approximations, which is equivalent to computing the largest Z- eigenvalues. As shown in [2,40], it is NP-hard to compute extreme eigenvalues of tensors. For nonnegative tensors, the largest H-eigenvalues can be computed by using Perron Frobenius theorem [4,23]. 23

3 Real eigenvalues of nonsymmetric tensors 3 For nonsymmetric tensors, there is no much earlier work on computing all real Z/Heigenvalues. An elementary approach for this task is to solve the polynomial systems (.3) and (.4) directly, for getting all complex solutions by symbolic methods, and then choose real ones. This approach is typically very expensive to be used in computation. A numerical experiment for this is given in Sect There are fundamental differences between symmetric and nonsymmetric tensor eigenvalues. For symmetric tensors, the Eqs. (.3) and (.4) follow from the Karush Kuhn Tucker (KKT) conditions of some polynomial optimization problems, which show the existence of real Z-eigenvalues and H-eigenvalues. Furthermore, every symmetric tensor has finitely many Z-eigenvalues [5]. However, the Eqs. (.3) and (.4) for nonsymmetric tensors may not be the KKT conditions of some optimization problems. The following examples show the differences between symmetric and nonsymmetric tensor eigenvalues. Example.3 Consider the tensor A T 4 (R 2 ) such that A ijkl = 0 except A 2 = A 222 =, A 2 = A 222 =. By the definition, (λ, x) is a Z-eigenpair if and only if (x 2 + x2 2 )x 2 = λx, (x 2 + x2 2 )x = λx 2, x 2 + x2 2 =. One can check that A has no real Z-eigenvalues and neither complex ones. This is confirmed by Theorem 3.(i), because the semidefinite relaxation (3.4) is infeasible for the order k = 3. By the definition, (λ, x) is an H-eigenpair if and only if it satisfies (x 2 + x2 2 )x 2 = λx 3, (x 2 + x2 2 )x = λx2 3, (x, x 2 ) = (0, 0). The tensor A has no real H-eigenvalues. This is also confirmed by Theorem 4.2(i), because the semidefinite relaxation (4.5) is infeasible for the order k = 4. It is possible that a tensor has infinitely many Z-eigenvalues. Example.4 Consider the tensor A T 4 (R 2 ) such that A ijkl = 0 except A = A 22 =. Then, (λ, x) is a Z-eigenpair of A if and only if x 3 = λx, x 2 x 2 = λx 2, (.5) x 2 + x2 2 =. 23

4 4 J. Nie, X. Zhang One can check that every λ [0, ] is a real Z-eigenvalue, associated to Z-eigenvectors (± λ, ± λ). Note that the set of Z/H-eigenvalues of a nonsymmetric tensor is different with different definitions on Ax m. To clear this fact, we consider (2)-mode [Ax m ] (2) defined by ( [Ax m ] (2) := i,i 3,...,i m n A i ji 3...i m x i x i3 x im ) j=,...,n. By directly computation, there is only Z-eigenvalue and H-eigenvalue with (2)- mode for tensor A in Examples.3 and.4. These are different from Examples.3 and.4. However, we would like to remark that the above examples are not generic cases. That is, every generic tensor has finitely many Z-eigenvalues, and its number can be given by explicit formula. Here, the meaning of the word generic is in the sense of Zariski topology (i.e., there exists a polynomial φ in the entries of A such that if φ(a) = 0 then A has finitely many Z-eigenvalues). We refer this result to [3]. On the other hand, every tensor has finitely many H-eigenvalues, which is shown in Proposition 4.. From these facts, it is a well-posed question to compute all real Z/H-eigenvalues, for generic tensors. In this paper, we propose numerical methods for computing all real Z-eigenvalues (if there are finitely many ones) and all real H-eigenvalues. For symmetric tensors, the Z-eigenvalues and H-eigenvalues are critical values of some polynomial optimization problems. This property was used very much in [5]. The method in [5] is based on Jacobian SDP relaxations [26], which are used for polynomial optimization. Indeed, the same kind of method can be used to compute all local minimum values of polynomial optimization [29]. However, the method in [5] is not suitable for computing eigenvalues of nonsymmetric tensors, because their eigenvalues are no longer critical values of polynomial optimization problems. This paper is organized as follows. Section 2 presents some preliminaries on polynomial optimization, tensor eigenvalues and an application example in higher order Markov chain. Section 3 proposes Lasserre type semidefinite relaxations for computing Z-eigenvalues. If there are finitely many real Z-eigenvalues, each of them can be computed by solving a finite sequence of semidefinite relaxations. Section 4 proposes Lasserre type semidefinite relaxations for computing all H-eigenvalues. Each of them can be computed by solving a finite sequence of semidefinite relaxations. Numerical examples are shown in Sect. 5. We make some discussions in Sect Preliminaries and motivation In this section, we recall some basics on polynomial optimization and tensor eigenvalues. After that, we present an application example arising in higher order Markov chain. 23

5 Real eigenvalues of nonsymmetric tensors 5 2. Preliminaries In this subsection, we review some basics in polynomial optimization. We refer to [7,8] for surveys in the area. In the space R n, u denotes the standard Euclidean norm of a vector u. Fort R, t denotes the smallest integer not smaller than t. Let R[x] be the ring of polynomials with real coefficients and in variables x := (x,...,x n ). The degree of a polynomial p, denoted by deg(p), is the total degree, i.e., the highest degree of its monomials. Denote by R[x] d the set of real polynomials in R[x] whose degrees are no greater than d. For a polynomial tuple h = (h, h 2,...,h s ), the ideal generated by h is the set I (h) := h R[x]+h 2 R[x]+ +h s R[x]. The k-th truncation of I (h) is the set I k (h) := h R[x] k deg(h ) + +h s R[x] k deg(hs ). The complex and real algebraic varieties of h are respectively defined as V C (h) := {x C n h(x) = 0}, V R (h) := V C (h) R n. A polynomial p is said to be sum of squares (SOS) if there exist p, p 2,...p r R[x] such that p = p 2 + p p2 r. The set of all SOS polynomials is denoted as [x]. For a given degree m, denote [x] m := [x] R[x] m. The quadratic module generated by a polynomial tuple g = (g,...,g t ) is Q(g) := [x]+g [x]+ +g t [x]. The k-th truncation of the quadratic module Q(g) is the set Q k (g) := [x] 2k + g [x] 2k deg(g ) + +g t [x] 2k deg(gt ). Clearly, it holds that I (h) = I k (h), Q(g) = Q k (g). k k If the tuple g is empty, then Q(g) = [x] and Q k (g) = [x] 2k. Let N be the set of nonnegative integers. For x := (x,...,x n ), α := (α,...,α n ) and a degree d, denote x α := x α xα n n, α :=α + +α n, N n d := {α Nn : α d}. 23

6 6 J. Nie, X. Zhang Denote by R Nn d the space of all vectors y that are indexed by α N n d.fory RNn d, we can write it as y = (y α ) α N n d. For f = α N n d f αx α R[x] d and y R Nn d, we define the operation f, y := f α y α. (2.) α N n d For every fixed f R[x] d, f, y is a linear functional in y; every fixed y R Nn d, f, y is a linear functional in f. The operation, gives a dual relationship between the polynomial space R[x] d and the vector space R Nn d.in f, y, the polynomial f is in the left side, while the vector y is in the right. For an integer t d and y R Nn d, denote the t-th truncation of y as y t := (y α ) α N n t. (2.2) Let q R[x] with deg(q) 2k. For each y R Nn 2k, qp 2, y is a quadratic form in vec(p), the coefficient vector of the polynomial p, with deg(qp 2 ) 2k. LetL q (k) (y) be the symmetric matrix such that qp 2, y =vec(p) T ( L (k) q (y) ) vec(p). (2.3) The matrix L q (k) (y) is called the k-th localizing matrix of q generated by y. It is linear in y. For instance, when n = 2, k = 2 and q = x x 2 x 2 x2 2, L (2) (y) = x x 2 x 2 x2 2 y y 20 y 02 y 2 y 30 y 2 y 2 y 2 y 03 y 2 y 30 y 2 y 3 y 40 y 22 y 22 y 3 y 3. y 2 y 2 y 03 y 22 y 3 y 3 y 3 y 22 y 04 If q = (q,...,q r ) is a tuple of polynomials, we then define L q (k) (y) := L (k) q (y)... L (k) q r (y). It is a block diagonal matrix. When q = (the constant polynomial), L (k) (y) is called the k-th moment matrix generated by y, and we denote For instance, when n = 2 and k = 2, 23 M k (y) := L (k) (y). (2.4)

7 Real eigenvalues of nonsymmetric tensors 7 y 00 y 0 y 0 y 20 y y 02 y 0 y 20 y y 30 y 2 y 2 M 2 (y) = y 0 y y 02 y 2 y 2 y 03 y 20 y 30 y 2 y 40 y 3 y 22. y y 2 y 2 y 3 y 22 y 3 y 02 y 2 y 03 y 22 y 3 y 04 For a degree d, denote the monomial vector [x] d := [, x,..., x n, x 2, x x 2,...,xn 2,..., ] T xm,...,xm n. (2.5) As shown in Example.4, there may be infinitely many real Z-eigenvalues for some tensors. But this is not the case for a generic tensor. In (.3), a real Z-eigenpair (λ, u) of a tensor A is called isolated if there exists ε>0 such that no other real Z-eigenpair (μ, v) satisfies λ μ + u v <ε. Similarly, a real Z-eigenvalue λ is called isolated if there exists ε>0 such that no other real Z-eigenvalue μ satisfies λ μ <ε. In practice, we need to check whether a Z-eigenvalue is isolated or not. Denote the function F(λ, x) := [ x T ] x Ax m. (2.6) λx Clearly, (λ, u) is a Z-eigenpair if and only if F(λ, u) = 0. Let J(λ, x) be the Jacobian matrix of the vector function F(λ, x) with respect to (λ, x). Lemma 2. Let A T m (R n ) and λ be a real Z-eigenvalue of A. (i) If u is a real Z-eigenvector associated to λ and J(λ, u) is nonsingular, then (λ, u) is isolated. (ii) If u,...,u N are the all real Z-eigenvectors of A associated to λ and each J(λ, u i ) is nonsingular, then λ is an isolated real Z-eigenvalue of A. Lemma 2. follows directly from the Inverse Function Theorem. For cleanness of the paper, its proof is omitted here. 2.2 Motivation: application in higher order Markov chain In higher order Markov chains [9,20], an mth order Markov chain fits observed data by an m-order nonsymmetric tensor P T m (R n ) whose entries are given such that P i i 2...i m = Prob(X t = i X t = i 2 ; ;X t m+ = i m ) [0, ], n P i i 2...i m =, i 2,...,i m {,...,n}. i = 23

8 8 J. Nie, X. Zhang Such P is called a transition probability tensor. A vector v R n is said to be a limiting (or stationary) probability distribution for P if Pv m = v, e T v =, v 0, where e is the vector of all ones. The existence and uniqueness of stationary probability distributions were discussed in [9]. Earlier work on this subject can be found in [3,9,20]. Interestingly, the stationary probability distributions can be obtained by computing the Z-eigenvalue of nonsymmetric tensor P. This can be seen as follows. Suppose v is a stationary probability distribution of P.Letu = v/ v, then u T u =, u 0, = e T v = v (e T u), v = u/(e T u), Pu m = v m Pvm = (e T u) m v = (e T u) m 2 u. That is, u is a Z-eigenvector of P associated to the Z-eigenvalue (e T u) m 2. Conversely, if λ is a Z-eigenvalue with the Z-eigenvector u 0, then Pu m = λu, u T u =, λ(e T u) = e T (Pu m ) = i 2,...,i m n u i2 u im i,i 2,...,i m n n P i i 2...i m = i = So, λ = (e T u) m 2.Ifweletv = Pv m = u e T u, then P i i 2...i m u i2 u im = i 2,...,i m n u i2 u im = (e T u) m. λu (e T u) m Pum = (e T u) m = v, et v =, v 0. That is, v is a stationary probability distribution vector. The above can be summarized as follows: v v is a nonneg- If v is a stationary probability distribution vector for P, then u = ative Z-eigenvector. If u is a nonnegative Z-eigenvector for P, then v = distribution vector. u is a stationary probability e T u Therefore, the stationary probability distribution vectors can be obtained by computing all nonnegative Z-eigenvectors. 3 Computing real Z-eigenvalues In this section, we first reformulate Z-eigenvalue computation as polynomial optimization problem. 23

9 Real eigenvalues of nonsymmetric tensors 9 Let A T m (R n ) be a tensor. A real pair (λ, u) is a Z-eigenpair of A if and only if Au m = λu and u T u =. So, Hence, u is a Z-eigenvector if and only if λ = λu T u = u T Au m = Au m. Au m = (Au m )u, u T u =, and the associated Z-eigenvalue is Au m. A generic tensor has finitely many Z- eigenvalues that are all simple, as shown in [3]. For special tensors, there might be infinitely many ones (see Example.4). In this section, we aim at computing all real Z-eigenvalues when there are finitely many ones. Let h be the polynomial tuple: h = ( Ax m (Ax m )x, x T x ). (3.) Then, u is a Z-eigenvector of A if and only if h(u) = 0. Let Z(R, A) denote the set of Z-eigenvalues of A. If it is a finite set, we list Z(R, A) monotonically as λ <λ 2 < <λ N. We aim at computing them sequentially, from the smallest to the largest. 3. The smallest Z-eigenvalue To compute the smallest Z-eigenvalue λ, we consider the polynomial optimization problem min f (x) := Ax m s.t. h(x) = 0, (3.2) where h is as in (3.). Note that u is a Z-eigenvector if and only if h(u) = 0, with the Z-eigenvalue f (u). The optimal value of (3.2) isλ,ifv R (h) =.Let k 0 = (m + )/2. (3.3) Lasserre s hierarchy [6] of semidefinite relaxations for solving (3.2) is f,k := min f, y s.t., y =, L (k) h (y) = 0, M k (y) 0, y R Nn 2k, (3.4) for the orders k = k 0, k 0 +,... See (2.3) (2.4) for the notation L (k) h (y) and M k(y). In the above, X 0 means that the symmetric matrix X is positive semidefinite. The dual optimization problem of (3.4) is 23

10 0 J. Nie, X. Zhang { f 2,k := max γ s.t. f γ I 2k (h) + [x] 2k. (3.5) As in [6], it can be shown that for all k f 2,k f,k λ and the sequences { f,k } and { f 2,k } are monotonically increasing. Theorem 3. Let A T m (R n ) and let Z(R, A) be the set of its real Z-eigenvalues. Then, we have the properties: (i) The set Z(R, A) = if and only if the semidefinite relaxation (3.4) is infeasible for some order k. (ii) If Z(R, A) = and λ is the smallest real Z-eigenvalue, then lim f 2,k k = lim f,k k = λ. (3.6) If, in addition, Z(R, A) is a finite set, then for all k sufficiently big f 2,k = f,k = λ. (3.7) (iii) Let k 0 be as in (3.3). Suppose y is a minimizer of (3.4). If there exists t ksuch that rank M t k0 (y ) = rank M t (y ), (3.8) then λ = f,k and there are r := rank M t (y ) distinct real Z-eigenvectors u,...,u r associated to λ. (iv) Suppose Z(R, A) is a finite set. If there are finitely many real Z-eigenvectors associated to λ, then, for all k big enough and for every minimizer y of (3.4), the condition (3.8) is satisfied for some t k. Proof (i) We prove the equivalence in two directions. Sufficiency: Assume (3.4) is infeasible for some k. Then A has no real Z-eigenpairs. Suppose otherwise (λ, u) is such a one. Then [u] 2k [see (2.5) for the notation] is feasible for (3.4) for all values of k, which is a contradiction. So Z(R, A) =. Necessity: Assume Z(R, A) =. Then the equation h(x) = 0 has no real solutions. By the Positivstellensatz [2], I (h) + [x]. So, when k is big enough, I 2k (h)+ [x] 2k, and then the optimization (3.5) is unbounded from above. By duality theory, (3.4) must be infeasible, for all k big enough. (ii) Note that x T x is a polynomial in the tuple h. So, (x T x ) 2 I (h) and the set (x T x ) 2 0 is compact. The ideal I (h) is archimedean [6]. The asymptotic convergence (3.6) is proved in Theorem 4.2 of [6]. Next, we prove the finite convergence (3.7) when Z(R, A) = is a finite set. Write Z(R, A) ={λ,...,λ N }, with λ < <λ N.Letb,...,b N R[t] be the 23

11 Real eigenvalues of nonsymmetric tensors univariate real polynomials in t such that b i (λ j ) = 0 when i = j and b i (λ j ) = when i = j.fori =,...,N, let ( 2. s i := (λ i λ ) b i ( f (x))) Let s := s + +s N. Then, s [x] 2k for some k > 0. The polynomial ˆ f := f λ s vanishes identically on V R (h). By the Real Nullstellensatz [2, Corollary 4..8], there exist an integer l>0 and q [x] such that ˆ f 2l + q I (h). For all ε>0 and c > 0, we can write ˆ f + ε = φ ε + θ ε, where φ ε = cε 2l ( fˆ 2l + q), ( θ ε = ε + f ˆ/ε + c( f ˆ/ε) 2l) + cε 2l q. By Lemma 2. of [25], when c 2l, there exists k 2 k such that, for all ε>0, Hence, we can get φ ε I 2k2 (h), θ ε [x] 2k2. f (λ ε) = φ ε + σ ε, where σ ε = θ ε + s [x] 2k2 for all ε > 0. This implies that, for all ε > 0, γ = λ ε is feasible in (3.5) for the order k 2. Thus, we get f 2,k 2 λ. Note that f 2,k f,k λ for all k and the sequence { f 2,k } is monotonically increasing. So, (3.7) must be true when k k 2. (iii) Note that M t (y ) 0 and L (t) h (y ) = 0, because t k. When (3.8) is satisfied, by Theorem. of [6], there exist r := rank M t (y ) vectors u,...,u r V R (h) such that (we refer to (2.5) for the notation [u i ] 2t ) y 2t = c [u ] 2t + +c r [u r ] 2t, with numbers c,...,c r > 0. The condition, y = implies that c + +c r =. By the notation, as in (2.), we can see that f, [u i ] 2k = f (u i ),so f,k = f, y =c f (u ) + +c r f (u r ), 23

12 2 J. Nie, X. Zhang f,k f (u i ) i =,...,r, because each [u i ] 2k is a feasible point for (3.4). Thus, we must have f,k = f (u ) = = f (u r ). Also note that f,k λ and each u i is a Z-eigenvector. So, f,k = f (u ) = = f (u r ) = λ. (iv) When Z(R, A) is a finite set, both { f,k } and { f 2,k } have finite convergence to λ, by item (iii). If (3.2) has finitely many minimizers, i.e., λ has finitely many Z-eigenvectors, the rank condition (3.8) must be satisfied when k is sufficiently big. The conclusion follows from Theorem 2.6 of [27]. In computation, we start to solve the semidefinite relaxation (3.4) from the smallest value of k, which is k 0 as in (3.3). If (3.4) is infeasible for k = k 0, then A has no real Z-eigenvalues; if (3.4)isfeasiblefork = k 0, then solve it for an optimizer y.if y satisfies (3.8), then we get λ = f,k ; otherwise, increase the value of k by one, and then repeat the above process. See Algorithm 3.6 for the implementation details. The tensor A in Example.3 has no real Z-eigenvalues. This is confirmed by that the relaxation (3.4) is infeasible for k = 3. Remark 3.2 () The rank condition (3.8) can be used as a criterion to check whether f,k = λ or not. If it is satisfied, then we can get r distinct minimizers u,...,u r of (3.2), i.e., each u i is Z-eigenvector of A. The vectors u i can be computed by the method in [0], which is implemented in GloptiPoly 3 []. (2) Suppose (3.8) holds. If rank M k (y ) is maximum among the set of all optimizers of (3.4), then we can get all mimizers of (3.2) [8, 6.6], i.e., we can get all real Z-eigenvectors associated to λ. When (3.4) (3.5) are solved by primal-dual interior point methods, typically we can get all real Z-eigenvectors associated to λ, provided there are finitely many. (3) The condition (3.8) requires to know the ranks of M t k0 (y ) and M t (y ).Itwould be hard to evaluate them when the matrices are singular or close to be singular. The matrix rank is equal to the number of positive singular values. In common practice, people often determine the rank by counting the number of singular values bigger than a tolerance (say, 0 6 ). This is a classical problem in numerical linear algebra. We refer to Demmel s book [8] for this question. (4) For generic polynomial optimization problems, Lasserre s hierarchies of semidefinite relaxations have finite convergence, as shown in [28]. However, the problem (3.2) is not generic, because h depends on f. So, the finite convergence of (3.4) cannot be proved by using the results of [28]. 23

13 Real eigenvalues of nonsymmetric tensors Larger Z-eigenvalues Suppose the ith smallest Z-eigenvalue λ i is known. We want to determine whether the next larger one λ i+ exists or not. If it exists, we show how to compute it; if it does not, we give a certificate for the nonexistence. Let δ>0 be a small number. Consider the problem min f (x) s.t. h(x) = 0, f (x) λ i + δ. (3.9) Clearly, the optimal value of (3.9) is the smallest Z-eigenvalue that is greater than or equal to λ i + δ. Let ω(λ i + δ) := min{λ Z(R, A) : λ λ i + δ}. (3.0) Lasserre s hierarchy of semidefinite relaxations for solving (3.9) is f,k i+ := min f, y s.t., y =, L (k) h (y) = 0, M k (y) 0, L (k) f λ i δ (y) 0, y RNn 2k, (3.) for the orders k = k 0, k 0 +,... The dual problem of (3.) is { f 2,k i+ := max γ s.t. f γ I 2k (h) + Q k ( f λ i δ). (3.2) Similar to Theorem 3., we have the following convergence result. Theorem 3.3 Let A T m (R n ) and Z(R, A) be the set of real Z-eigenvalues of A. Suppose λ i Z(R, A). Then, we have the properties: (i) The intersection Z(R, A) [λ i + δ, + ) = if and only if the semidefinite relaxation (3.) is infeasible for some order k. (ii) If Z(R, A) [λ i + δ, + ) =, then lim f 2,k k i+ = lim f,k k i+ = ω(λ i + δ). (3.3) If, in addition, Z(R, A) [λ i + δ, + ) is a finite set, then f 2,k i+ = f,k i+ = ω(λ i + δ) (3.4) for all k sufficiently big. (iii) Suppose y is a minimizer of (3.). If (3.8) is satisfied for some t k, then ω(λ i + δ) = f,k i+ and there are r := rank M t(y ) distinct real Z-eigenvectors u,...,u r, associated to ω(λ i + δ). (iv) Suppose Z(R, A) [λ i + δ, + ) is a finite set and ω(λ i + δ) has finitely many real Z-eigenvectors. Then, for all k big enough, and for every minimizer y of (3.), there exists t k satisfying (3.8). 23

14 4 J. Nie, X. Zhang Proof The proof is mostly the same as for Theorem 3.. In the following, we only list the major differences. (i) Necessity: If Z(R, A) [λ i + δ, + ) =, then (3.9) is infeasible. By the Positivstellensatz, I (h) + Q( f λ i δ). The rest of the proof is the same as for Theorem 3.(i). (ii) The ideal I (h) is archimedean, and so is I (h) + Q( f λ i δ). The asymptotic convergence (3.3) can be implied by Theorem 4.2 of [6]. To prove the finite convergence (3.4), we follow the same proof as for Theorem 3.(ii). Suppose Z(R, A) [λ i + δ, + ) ={ν,...,ν L }. Construct the polynomial s same as there and let Then ˆ f vanishes identically on the set ˆ f = f ω(λ i + δ) s. {x R n : h(x) = 0, f (x) λ i δ 0}. By the Real Nullstellensatz [2, Corollary 4..8], there exist an integer l>0and q Q( f λ i δ) such that fˆ 2l + q I (h). The rest of the proof is the same, except replacing [x] by Q( f λ i δ), and [x] 2k by Q k ( f λ i δ). (iii) (iv) The proof is the same as for Theorem 3.(iii) (iv). The convergence of semidefinite relaxations (3.) (3.2) can be checked by the condition (3.8). When it is satisfied, the real Z-eigenvectors u,...,u r can be computed by the method in [0]. Typically, we can get all real Z-eigenvectors if primal-dual interior-point methods are used to solve the semidefinite relaxations. We refer to Remark 3.2. Next, we show how to use ω(λ i + δ) to determine λ i+. Assume λ i is isolated (otherwise, there are infinitely many Z-eigenvalues). If λ i is the largest real Z-eigenvalue, then λ,...,λ i are the all real Z-eigenvalues and we can stop; otherwise, the next larger one λ i+ exists. For such case, if δ>0in(3.9) is small enough, then ω(λ i + δ) as in (3.0) equals λ i+. Consider the optimization problem { ν + (λ i,δ):= max f (x) s.t. h(x) = 0, f (x) λ i + δ. (3.5) The optimal value of (3.5) is the largest Z-eigenvalue of A that is smaller than or equal to λ i + δ, i.e., ν + (λ i,δ)= max{λ Z(R, A) : λ λ i + δ}. The next larger Z-eigenvalue λ i+ can be determined as follows. 23

15 Real eigenvalues of nonsymmetric tensors 5 Theorem 3.4 Let A T m (R n ) and δ>0. Assume λ i is an isolated Z-eigenvalue of A and λ max is the largest one. Then, we have the properties: (i) For all δ>0 sufficiently small, ν + (λ i,δ)= λ i. (ii) If ν + (λ i,δ) = λ i and (3.) is infeasible for some k, then λ i = λ max and the next larger Z-eigenvalue λ i+ does not exist. (iii) If ν + (λ i,δ) = λ i and the condition (3.8) is satisfied for some k, then the next larger Z-eigenvalue λ i+ = f,k i+. Proof (i) Note that ν + (λ i,δ) is the smallest Z-eigenvalue greater than or equal to λ i +δ. When λ i is isolated, for δ>0 sufficiently small, we must have ν + (λ i,δ)= λ i. (ii) When (3.) is infeasible for some k, by Theorem 3.3(i), all the Z-eigenvalues are smaller than λ i + δ. Note that ν + (λ i,δ)is the largest Z-eigenvalue that is smaller than or equal to λ i + δ. Ifλ i = ν + (λ i,δ), then λ i must be the largest Z-eigenvalue, i.e., λ i = λ max. (iii) When (3.8) is satisfied for some k, by Theorem 3.3(iii), we know ω(λ i + δ) = f,k i+. Note that ν + (λ i,δ)is the largest Z-eigenvalue that is smaller than or equal to λ i + δ, while ω(λ i + δ) is the smallest Z-eigenvalue that is bigger than or equal to λ i + δ. Since λ i = ν + (λ i,δ)and λ i is isolated, we must have λ i+ = ω(λ i + δ), which is the smallest Z-eigenvalue bigger than λ i. In the next, we show how to check if a real Z-eigenvalue λ i is isolated or not. Lemma 2. can be used to verify the isolatedness. It only gives a sufficient condition, which may not be necessary sometimes. Here, we give a sufficient and necessary condition. Consider the optimization problem { ν (λ i,δ):= min f (x) s.t. h(x) = 0, f (x) λ i δ. (3.6) Clearly, for all δ>0, it holds that ν (λ i,δ) λ i ν + (λ i,δ). Lemma 3.5 Let ν + (λ i,δ), ν (λ i,δ) be the optimal values as in (3.5) and (3.6). Then, λ i is an isolated real Z-eigenvalue of A if and only if for some δ>0 ν + (λ i,δ)= ν (λ i,δ). (3.7) When the above holds, λ i is the unique Z-eigenvalue of A in [λ i δ, λ i + δ]. 23

16 6 J. Nie, X. Zhang Proof By the construction of h as in (3.), u is a Z-eigenvector if and only if h(u) = 0. So, ν + (λ i,δ)is the largest Z-eigenvalue that is smaller than or equal to λ i + δ, while ν (λ i,δ) is the smallest one that is greater than or equal to λ i δ. We prove the equivalence in two directions. Necessity: If λ i is isolated, then for δ > 0 small enough, λ i is the unique Z- eigenvalue of A in the interval [λ i δ, λ i + δ]. So, (3.7) holds. Sufficiency: Assume (3.7) holds for some δ>0, then ν (λ i,δ)= λ i = ν + (λ i,δ). So, λ i is the unique Z-eigenvalue in [λ i δ, λ i + δ], and it must be isolated. The problems (3.5) and (3.6) are polynomial optimization. Their optimal values ν + (λ i,δ), ν (λ i,δ)can be computed by solving Lasserre type semidefinite relaxations that are similar to (3.) (3.2). 3.3 An algorithm for computing real Z-eigenvalues For a given tensor A, we compute its real Z-eigenvalues if they exist, from the smallest to the largest. First, we compute λ if it exists, by solving (3.4) (3.5). After getting λ, we solve (3.) (3.2) and then determine λ 2.Ifλ 2 does not exist, we stop; otherwise, we then determine λ 3. Repeating this procedure, we can get all the real Z-eigenvalues, when there are finitely many ones. This results in the following algorithm. Algorithm 3.6 For a given tensor A T m (R n ), compute its real Z-eigenvalues as follows: Step 0: Let k := k 0, with k 0 as in (3.3). Step : Solve the relaxation (3.4). If it is infeasible, then A has no real Z-eigenvalues and stop; if it is feasible, compute an optimizer y. Step 2: If (3.8) is satisfied, then λ = f,k and go to Step 3 with i =. Otherwise, let k := k + and go to Step. Step 3: Let δ = Solve (3.5), (3.6) for the optimal values ν + (λ i,δ), ν (λ i,δ). If ν + (λ i,δ)= ν (λ i,δ), then λ i is an isolated Z-eigenvalue and go to Step 4. Otherwise, let δ := δ/5 and compute ν + (λ i,δ), ν (λ i,δ)again. Repeat this process until we get ν + (λ i,δ)= ν (λ i,δ). Step 4: Let k := k 0, with k 0 as in (3.3). Step 5: Solve the relaxation (3.). If it is infeasible, the largest Z-eigenvalue is λ i and stop. Otherwise, compute an optimizer y for it. Step 6: If (3.8) is satisfied, then λ i+ = f,k i+ and go to Step 3 with i := i +. Otherwise, let k := k + and go to Step 5. The correctness of Algorithm 3.6 and its convergence properties are proved in Theorems 3., 3.3 and Computing real H-eigenvalues In this section, we compute real H-eigenvalues. For every tensor A, the number of H-eigenvalues is always finite. In Definition.2, if λ, u are allowed to achieve com- 23

17 Real eigenvalues of nonsymmetric tensors 7 plex values, then we call such λ a complex H-eigenvalue and such u a complex H-eigenvector. Proposition 4. Every tensor A T m (C n ) has n(m ) n complex H-eigenvalues, counting their multiplicities. Proof Let I T m (C n ) be the identity tensor whose only non-zero entries are I ii...i = fori =, 2,...,n. Note that λ is a complex H-eigenvalue if and only if there exists 0 = u C n such that Au m = λu [m ], that is, (A λi)u m = 0. By the definition of resultant [39], which we denote by Res, λ is an H-eigenvalue if and only if Res ( (A λi)x m ) = 0. (4.) The resultant Res ( (A λi)x m ) is homogeneous in the entries of A and λ. Ithas degree D := n(m ) n. We can expand it as Res ( (A λi)x m ) = p 0 (A) + p (A)λ + + p D (A)λ D. By the homogeneity of Res, we know p D (A) = Res ( Ix m ) = 0, because the homogeneous polynomial system Ix m = 0 has no nonzero complex solutions. Hence, the leading coefficient of the polynomial Res ( (A λi)x m ) in λ is not zero, and the degree is D. This implies that (4.) has D complex solutions, counting multiplicities, and the lemma is proved. Recall that (λ, u) is a real H-eigenpair if and only if Au m = λu [m ], 0 = u R n. Let m 0 be the largest even number less than or equal to m, i.e., m 0 = 2 (m )/2. Note that m m 0 m. We can normalize u as (u ) m 0 + +(u n ) m 0 =. (4.2) Under this normalization, the H-eigenvalue λ can be given as Let h be the polynomial tuple h := λ = λ(u [m 0 m+] ) T u [m ] = (u [m 0 m+] ) T Au m. ( Ax m ( (x [m 0 m+] ) T Ax m ) x [m ], n ) (x i ) m 0. (4.3) i= 23

18 8 J. Nie, X. Zhang Then, u is an H-eigenvector normalized as in (4.2) if and only if h(u) = 0. Since A has finitely many H-eigenvalues, we can order the real ones monotonically as μ <μ 2 < <μ N, if at least one of them exists. We call μ i the ith smallest H-eigenvalue. 4. The smallest H-eigenvalue In this subsection, we show how to determine μ.leth be as in (4.3), then μ equals the optimal value of the optimization problem { min f (x) := (x [m 0 m+] ) T Ax m s.t. h(x) = 0. Lasserre s hierarchy [6] of semidefinite relaxations for solving (4.4) is ρ,k := min f, z s.t., z =, L (k) h (z) = 0, M k (z) 0, z R Nn 2k, (4.4) (4.5) for the orders k = k 0, k 0 +,...,where The dual optimization problem of (4.5) is k 0 := (m 0 + m )/2. (4.6) { ρ 2,k := max γ s.t. f γ I 2k (h) + [x] 2k. (4.7) As can be shown in [6], ρ 2,k ρ,k μ for all k, and the sequences {ρ,k } are monotonically increasing. {ρ 2,k } and Theorem 4.2 Let A T m (R n ) and H(R, A) be the set of its real H-eigenvalues. Then, we have the properties: (i) The set H(R, A) = if and only if the semidefinite relaxation (4.5) is infeasible for some order k. (ii) If H(R, A) =, then for all k sufficiently big ρ,k = ρ 2,k = μ. (4.8) (iii) Let k 0 be as in (4.6). Suppose z is a minimizer of (4.5). If there exists an integer t k such that rank M t k0 (z ) = rank M t (z ), (4.9) 23

19 Real eigenvalues of nonsymmetric tensors 9 then μ = ρ,k and there are r := rank M t (z ) distinct real H-eigenvectors v,...,v r associated to μ and normalized as in (4.2). (iv) Suppose A has finitely many real H-eigenvectors associated to μ. Then, for all k big enough and for every minimizer z of (4.5), there exists an integer t k satisfying (4.9). Proof It can be proved in the same way as for Theorem 3.. The only difference is that H(R, A) is always a finite set, by Proposition 4.. To avoid being repetitive, the proof is omitted here. The tensor A in Example.3 has no real H-eigenvalues. This can be confirmed by Theorem 4.2(i), because the semidefinite relaxation (4.5) is infeasible for the order k = 4. The rank condition (4.9) is a criterion for checking the finite convergence in (4.8). When it is satisfied, the real H-eigenvectors u,...,u r can be computed by the method in [0]. Typically, we can get all H-eigenvectors if primal-dual interior-point methods are used to solve (4.5). We refer to Remark Larger H-eigenvalues Suppose the ith smallest H-eigenvalue μ i is known. We want to determine the next larger one μ i+. If it exists, we show how to compute it; if not, we get a certificate for the nonexistence. Let δ>0 be a small number. Consider the optimization problem min f (x) s.t. h(x) = 0, f (x) μ i + δ, (4.0) where f, h are same as in (4.4). The optimal value of (4.0)isthesmallestH-eigenvalue of A that is greater than or equal to μ i + δ. Denote ϖ(μ i + δ) := min{μ H(R, A) : μ μ i + δ}. (4.) Lasserre s hierarchy of semidefinite relaxations for solving (4.0) is ρ,k i+ := min f, z s.t., z =, L (k) h (z) = 0, M k (z) 0, L (k) f μ i δ (z) 0, z RNn 2k, (4.2) for the orders k = k 0, k 0 +,... The dual problem of (4.2) is { ρ 2,k i+ := max γ s.t. f γ I 2k (h) + Q k ( f μ i δ). (4.3) The properties of relaxations (4.2) (4.3) are as follows. Theorem 4.3 Let A T m (R n ) and H(R, A) be the set of its real H-eigenvalues. Assume that μ i H(R, A). Then, we have the properties: 23

20 20 J. Nie, X. Zhang (i) The intersection H(R, A) [μ i + δ, + ) = if and only if the semidefinite relaxation (4.2) is infeasible for some order k. (ii) If H(R, A) [μ i + δ, + ) =, then for all k sufficiently big ρ 2,k i+ = ρ,k i+ = ϖ(μ i + δ). (4.4) (iii) Let z be a minimizer of (4.2). If (4.9) is satisfied for some t k, then there exists r := rank M t (z ) real H-eigenvectors v,...,v r that are associated to ϖ(μ i + δ) and that are normalized as in (4.2). (iv) Suppose A has finitely many real H-eigenvectors that are associated to ϖ(μ i +δ) and that are normalized as in (4.2). Then, for all k big enough and for all minimizer z of (4.2), there exists t k satisfying (4.9). Proof It can be proved in the same way as for Theorem 3.3. Note that H(R, A) is always a finite set, by Proposition 4.. In the following, we show how to use ϖ(μ i + δ) to determine μ i+. Consider the maximization problem { υi := max f (x) s.t. h(x) = 0, f (x) μ i + δ. (4.5) The optimal value of (4.5) is the largest H-eigenvalue of A that is smaller than or equal to μ i + δ, i.e., υ i = max{μ H(R, A) : μ μ i + δ}. The next larger H-eigenvalue μ i+ can be determined as follows. Theorem 4.4 Let A T m (R n ) and δ>0. Suppose μ i H(R, A) and μ max is the maximum real H-eigenvalue. Let ϖ(λ i + δ) be as in (4.). Then, we have: (i) For all δ>0 small enough, υ i = μ i. (ii) If υ i = μ i and (4.2) is infeasible for some k, then μ i = μ max. (iii) If υ i = μ i and (4.9) is satisfied for some k, then μ i+ = ρ,k i+. Proof The proof is the same as for Theorem 3.4. Note that A has finitely many H- eigenvalues, and μ i is always isolated, as in Proposition 4.. Since (4.5) is a polynomial optimization, the optimal value υ i can also be computed by solving Lasserre type semidefinite relaxations that are similar to (4.2) (4.3). 4.3 An algorithm for all H-eigenvalues We can compute all real H-eigenvalues of a tensor A sequentially, from the smallest one to the largest one, if they exist. A similar version of Algorithm 3.6 can be applied. 23

21 Real eigenvalues of nonsymmetric tensors 2 Algorithm 4.5 For a given tenosr A T m (R n ), compute its real H-eigenvalues as follows: Step 0: Let k = k 0, with k 0 as in (4.6). Step : Solve the relaxation (4.5). If it is infeasible, then A has no real H-eigenvalues and stop. Otherwise, compute an optimizer z. Step 2: If (4.9) is satisfied, then μ = ρ,k and go to Step 3 with i :=. Otherwise, let k := k + and go to Step. Step 3: Let δ = Solve (4.5) for its optimal value υ i.ifυ i = μ i,gotostep4. Otherwise, let δ := δ/5 and compute υ i again. Repeat this process until we get υ i = μ i. Step 4: Let k = k 0, with k 0 as in (4.6). Step 5: Solve the relaxation (4.2). If it is infeasible, the largest H-eigenvalue is μ i and stop. If it is feasible, compute an optimizer z. Step 6: If (4.9) is satisfied, then μ i+ = ρ,k i+ and go to Step 3 with i := i +. Otherwise, let i := i + and go to Step 5. The correctness of Algorithm 4.5 and its convergence properties are proved in Theorems 4.2, 4.3 and Numerical examples In this section, we present numerical experiments for computing real Z-eigenvalues and H-eigenvalues. The Algorithms 3.6 and 4.5 are implemented in MATLAB 7.0 on a Dell Desktop with Linux as OS with 8GB memory and Intel(R) CPU 2.8GHz. The software Gloptipoly 3 [] is used to solve the semidefinite relaxations in the algorithms. The computational results are displayed with four decimal digits, for cleanness of the presentation. The isolatedness of Z-eigenvalues are checked by Lemma 2. or Lemma 3.5. In checking conditions (3.8) and (4.9), the rank of a matrix is evaluated as the number of its singular values that are greater than 0 6. For odd order tensors, the Z-eigenvalues always appear in ± pairs, so only nonnegative Z- eigenvalues are displayed for them. In each table of this section, the time denotes the consuming time in seconds of the computation. 5. Numerical examples of Z(H)-eigenvalues Example 5. [9, Example 3] Consider the tensor A T 4 (R 2 ) with entries A i i 2 i 3 i 4 = 0 except A = 25., A 22 = 25.6, A 22 = 24.8, A 2222 = 23. Applying Algorithms 3.6 and 4.5, we get all the real Z/H-eigenvalues. They are shown in Table. It took about s to compute Z-eigenpairs, and about 3 s to compute H-eigenpairs. The convergence f,k i λ i and ρ,k i μ i occurs as follows: f,3 = λ, f,3 2 = λ 2, ρ,4 = μ,ρ,5 2 = μ 2,ρ,4 3 = μ 3. 23

22 22 J. Nie, X. Zhang Table Z/H-eigenpairs of the tensor in Example 5. Z-eigenvalue Z-eigenvector ± (0, ) ± (, 0) H-eigenvalue H-eigenvector ± (0, ) ± (, 0) ± (0.8527, ± ) The Z-eigenvalues λ,λ 2 are isolated. This can be verified by Lemma 3.5, because ν + (λ i, 0.05) = ν (λ i, 0.05) for all λ i. Their isolatedness can also be verified by Lemma 2.. For the function F(λ, u) as in (2.6), its Jacobian matrices at (λ, u ), (λ 2, u 2 ) are respectively , They are both nonsingular. By a direct calculation, one can show that the real Z- eigenvalues are λ = 23, λ 2 = 25., while the real H-eigenvalues are μ = 23, μ 2 = 25., μ 3 = ( )/20. The computed Z/H-eigenpairs are all correct, up to round-off errors. Actually, we have Au 3 i λ i u i = Av 3 i μ i v [3] i i =, 2 and Av 3 3 μ 3v [3] =0for Example 5.2 [33, Example ] A i i 2 i 3 = 0 except Consider the tensor A T 3 (R 3 ) with the entries A = , A 2 = , A 3 = 0.440, A 2 = 0.854, A 22 = 0.099, A 23 = , A 3 = , A 32 = 0.385, A 33 = , A 2 = , A 22 = , A 32 = , A 22 = 0.764, A 222 = , A 232 = , A 32 = , A 322 = , A 332 = 0.25, A 3 = 0.387, A 23 = , A 33 = 0.35, A 23 = 0.355, A 223 = , A 233 = , A 33 = 0.975, A 323 = , A 333 = By Algorithms 3.6 and 4.5, we get all the real Z-eigenvalues (only nonnegative ones are computed because the order is odd), and get all the real H-eigenvalues. Three Z- eigenpairs (λ i, u i ) and four H-eigenpairs (μ i,v i ) are obtained. The results are shown in Table 2. It took about 4 s to compute the Z-eigenvalues, and about 5 s to get the H-eigenvalues. The three Z-eigenvalues λ,λ 2,λ 3 are all isolated. This can be verified by Lemma 3.5, because 23 ν (λ i, 0, 05) = ν + (λ i, 0.05)

23 Real eigenvalues of nonsymmetric tensors 23 Table 2 Z/H-eigenpairs of the tensor in Example 5.2 Z-eig. λ i Z-eigvec. u i Au 2 i λ i u i (0.3736, 0.703, 0.98) (0.6666, , ) (0.4086, , 0.637) H-eig. μ i H-eigvec. v i Avi 2 μ i v [2] i.3586 (0.5795, , ) (0.6782, , ) ( , , ) (0.4956, 0.674, 0.609) for all λ i. The isolatedness is also confirmed by Lemma 2.. The smallest singular values of the Jacobian matrix of F(λ, x) at (λ i, u i ) are respectively 0.605, 0.706, They are all nonsingular. The computed eignpairs are all correct, up to tiny numerical errors. The residual errors Aui 2 λ i u i, Avi 2 μ i v [2] i are shown in Table 2. Example 5.3 [4, 4.] Consider the tensor A T 3 (R 3 ) with A i i 2 i 3 = 0 except A = , A 2 = 0.443, A 3 = 0.94, A 2 = 0.443, A 22 = , A 23 = 0.590, A 3 = 0.94, A 32 = , A 33 = 0.02, A 2 = 0.443, A 22 = , A 32 = , A 22 = , A 222 = 0.283, A 232 = , A 32 = 0.590, A 322 = , A 332 = , A 3 = 0.94, A 23 = 0.590, A 33 = 0.02, A 23 = , A 223 = , A 323 = , A 333 = A 233 = , A 33 = 0.02, By Algorithms 3.6 and 4.5, we get all the real Z-eigenvalues (only nonnegative ones are computed because the order is odd), and get all the real H-eigenvalues. They are shown in Table 3. It took about 4 s to compute the Z-eigenvalues, and about 3 s to get the H-eigenvalues. It can be verified that ν (λ i, 0.05) = ν + (λ i, 0.05) for all λ i. By Lemma 3.5, we know λ,λ 2 are isolated Z-eigenvalues. This is also confirmed by Lemma 2.. The smallest singular value of the Jacobian matrix of F(λ, x) as in (2.6)at(λ, u ), (λ 2, u 2 ) are respectively , (For λ 2, there are four Z-eigenvectors u 2. At each of them, the smallest singular value is the same.) They are all nonsingular. The computed eigenpairs (λ i, u i ) and (μ i,v i ) are correct, up to small numerical errors. Their residual errors are shown in Table 3. 23

24 24 J. Nie, X. Zhang Table 3 Z/H-eigenvalues of the tensor in Example 5.3 Z-eig. λ i Z-eigvec. u i Aui 2 λ i u i ( , 0.395, ) ( , , ) ( 0.2, 0.995, ) (0.820, , 0.323) (0.065, , ) H-eig. μ i H-eigvec. v i Avi 2 μ i v [2] i (0.7954, , ) (0.760, , ) Table 4 Z/H-eigenvalues of the tensor in Example 5.4 n Z-eig. λ i ( 0) time H-eig. μ i time None ,.664, , ; , , , , , ,.9260, 4.040, , , , , , 8.844, , , 7.458, ,.090 Example 5.4 [24, Example 3.9] Consider the tensor A T 3 (R n ) such that ( A i i 2 i 3 = tan i i i ) 3. 3 Applying Algorithms 3.6 and 4.5, we get all the real Z-eigenvalues (only nonnegative ones are computed because the order is odd), and get all the real H-eigenvalues. The results are shown in Table 4, for the dimensions n = 2, 3, 4, 5. There are no real H-eigenvalues when n = 2. The Z-eigenvalues are all isolated. This is verified by Lemma 3.5, because ν (λ i, 0.05) = ν + (λ i, 0, 05) for all λ i. Since there are 28 eigenvalues, the eigenvectors are not shown, for neatness of the paper. They are all correct, up to small numerical errors. Example 5.5 Consider the tensor A T 4 (R 3 ) such that A i...i 4 = arctan(i i 2 2 i 3 3 i 4 4 ). By Algorithms 3.6 and 4.5, we get all the real Z/H-eigenvalues. It took about 3 s to get the Z-eigenvalues, and about 5 s to get the H-eigenvalues. The results are shown in Table 5. All the Z-eigenvalues λ i are isolated, which is verified by Lemma 3.5 because 23

25 Real eigenvalues of nonsymmetric tensors 25 Table 5 Z/H-eigenvalues of the tensor in Example 5.5 Z-eig. λ i Z-eigvec. u i Aui 3 λ i u i (0.9306, , 0.364) (0.2402, , ) (0.570, , ) H-eig. μ i H-eigvec. v i Aui 3 λ i u [3] i (0.943, , ) (0.2924, 0.939, ) (0.7566, , ) Table 6 Z/H-eigenvalues of the tensor in Example 5.6 Z-eig. λ i Z-eigvec. u i Aui 3 λ i u i (0.23, , ) (0.6244, 0.938, ) (0.6083, , ) H-eig. μ i H-eigvec. v i Avi 3 μ i v [3] i (0.2552, , 0.773) ( , , 0.897) (0.7732, , ) ν (λ i, 0.05) = ν + (λ i, 0.05) for all λ i. The computed Z/H-eigenpairs (λ i, u i ), (μ i,v i ) are all correct, up to small numerical errors. The residual errors are shown in Table 5. Example 5.6 Consider the tensor A T 4 (R 3 ) such that A i...i 4 = ( + i + 2i 2 + 3i 3 + 4i 4 ). By Algorithms 3.6 and 4.5, we get all the real Z/H-eigenvalues. It took about 4 s to compute Z-eigenvalues, and about 5 s for H-eigenvalues. The results are shown in Table 6.TheZ-eigenvalues λ i are all isolated. This is verified by Lemma 3.5, because ν (λ i, 0.000) = ν + (λ i, 0.000) for all λ i. The computed eigenpairs (λ i, u i ), (μ i,v i ) are all correct, up to small numerical errors. The residual errors are shown in Table 6. Example 5.7 Consider the tensor A T 5 (R n ) such that ( 5. A i...i 5 = ( ) j exp(i j )) j= 23

26 26 J. Nie, X. Zhang Table 7 Z/H-eigenvalues of the tensor in Example 5.7 n Z-eig. λ i ( 0) Time H-eig. μ i Time , , , , , , , 2.399, , , , , , , , 8, , , 4.473, , , , 5.036, 5.789, , , , Table 8 Z/H-eigenvalues of the tensor in Example 5.8 n Z-eig. λ i ( 0) time H-eig. μ i time , , , , 0.06, , , , , 0.568, ,.4958, , 0.007, , , , 0.000, , 0.002, , ,.576, , 4.089, , For the dimensions n = 2, 3, 4, 5, all the real Z/H-eigenvalues are found by Algorithms 3.6 and 4.5. Because the order is odd, only nonnegative Z-eigenvalues are computed. The results are shown in Table 7. The Z-eigenvalues are all isolated, verified by Lemma 3.5. This is because ν (λ i, 0.05) = ν + (λ i, 0.05) for all λ i. The computed eigenpairs are all correct, up to round-off errors. Since there are 34 computed eigenpairs, for neatness of the paper, the eigenvectors and the residual errors are not displayed. Example 5.8 Consider the tensor A T 3 (R n ) such that A i i 2 i 3 = ( ) i + 2i 2 + 3i 3 i i i 3 2. For the dimensions n = 2, 3, 4, we get all the real Z-eigenvalues (only nonnegative ones are computed because the order is odd), and get all the real H-eigenvalues. The results are shown in Table 8. 23

27 Real eigenvalues of nonsymmetric tensors 27 The Z-eigenvalues are all isolated, verified by Lemma 3.5. This is because ν (λ i, 0.00) = ν + (λ i, 0.00) for all λ i. The computed eigenpairs are all correct, up to round-off errors. Since there are 32 computed eigenpairs, for neatness of the paper, the eigenvectors and the residual errors are not displayed. 5.2 Numerical examples of stationary probability distribution vectors In this subsection, we report numerical examples on computing all stationary probability distribution vectors. Example 5.9 Consider the 3rd order 3-dimensional transition probability tensor P T 3 (R 3 ) whose nonzero entries are P 2 =, P 2 =, P 3 =, P 3 =, P 2 =, P 222 =, P 223 =, P 232 =, P 333 =. Its stationary probability distribution vectors are exactly v = (0,, 0), v 2 = (0, 0, ), v 3 = ( 2, 2, 0). By Algorithm 3.3, we get all the stationary probability distribution vectors, with the relaxation orders 2, 2, 3, respectively. It took about 4 s. One can verify that Pv 2 i v i =0fori =, 2, 3. In the following examples, for P T m (R n ), P ij... (i, j,... {,...,n}) denotes the matrix whose entries are given as (P ij... ) t t 2 = P t t 2 ij... ( t, t 2 n). Example 5.0 [] Consider the transition probability tensor P T 3 (R 4 ) that is given as: P = ; P 2 = ; P 3 = ; P 4 = It has only one nonnegative Z-eigenvalue. By Algorithm 3.6, we get the Z-eigenvector u = (0.477, 0.576, , ). 23