Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations

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1 Characterizing Real-Value Multivariate Complex Polynomials an Their Symmetric Tensor Representations Bo JIANG Zhening LI Shuzhong ZHANG December 31, 2014 Abstract In this paper we stuy multivariate polynomial functions in complex variables an the corresponing associate symmetric tensor representations. The focus is on fining conitions uner which such complex polynomials/tensors always take real values. We introuce the notion of symmetric conjugate forms an general conjugate forms, an present characteristic conitions for such complex polynomials to be real-value. As applications of our results, we iscuss the relation between nonnegative polynomials an sums of squares in the context of complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for complex tensors are introuce, extening properties from the Hermitian matrices. Finally, we iscuss an important property for symmetric tensors, which states that the largest absolute value of eigenvalue of a symmetric real tensor is equal to its largest singular value; the result is known as Banach s theorem. We show that a similar result hols in the complex case as well. Keywors: symmetric complex tensor, conjugate complex polynomial, tensor eigenvalue, tensor eigenvector, nonnegative complex polynomial, Banach s theorem. Mathematics Subject Classification: 15A69, 15A18, 15B57, 15B48. Research Center for Management Science an Data Analytics, School of Information Management an Engineering, Shanghai University of Finance an Economics, Shanghai , China. isyebojiang@gmail.com. Research of this author was supporte in part by National Natural Science Founation of China (Grant ) an National Science Founation (Grant CMMI ). Department of Mathematics, University of Portsmouth, Portsmouth PO1 3HF, Unite Kingom. zheningli@gmail.com. Research of this author was supporte in part by the Faculty of Technology RDI 2014, University of Portsmouth. Department of Inustrial an Systems Engineering, University of Minnesota, Minneapolis, MN 55455, USA. zhangs@umn.eu. Research of this author was supporte in part by National Science Founation (Grant CMMI ). 1

2 1 Introuction In this paper we set out to stuy the functions in multivariate complex variables which however always take real values. Such functions are frequently encountere in engineering applications arising from signal processing [2, 19], electrical engineering [27], an control theory [30]. It is interesting to note that such complex functions are usually not stuie by conventional complex analysis, since they are typically not even analytic because the Cauchy-Riemann conitions will never be satisfie unless the function in question is trivial. There has been a surge of research attention to solve optimization moels relate to such kin of complex functions [2, 27, 28, 13, 14]. Sorber et al. [29] evelope a MATLAB toolbox calle Complex Optimization Toolbox for optimization problems in complex variables, where the complex function in question is either preassume to be always real-value [27], or it is the moulus/norm of a complex function [2, 28]. An interesting question thus arises: Can such real-value complex functions be characterize? Inee there oes exist a class of special complex functions that always take real values: the Hermitian quaratic form x H Ax where A is a Hermitian matrix. In this case, the quaratic structure plays a key role. This motivates us to search more general complex polynomial functions with the same property. Interestingly, such complex polynomials can be completely characterize, as we will present in this paper. As is well-known, polynomials can be represente by tensors. The same question can be aske about complex tensors. In fact, there is a consierable amount of recent research attention on the applications of complex tensor optimization. For instance, Hilling an Suberythe [12] formulate a quantum entanglement problem as a complex multilinear form optimization uner the spherical constraint, an Zhang an Qi [33] iscusse a quantum eigenvalue problem, which arise from the geometric measure of entanglement of a multipartite symmetric pure state in the complex tensor space. Examples of complex polynomial optimization inclue Aittomaki an Koivunen [1] who formulate the problem of beam-pattern synthesis in array signal processing as complex quartic polynomial minimization, an Aubry et al. [2] who moele a raar signal processing problem by complex polynomial optimization. Solution methos for complex polynomial optimization can be foun in, e.g. [27, 13, 14]. As mentione before, polynomials an tensors are known to be relate. In particular in the real omain, homogeneous polynomials (or forms) are bijectively relate to symmetric tensors (aka super-symmetric in some papers in the literature), i.e., the components of the tensor is invariant uner the permutation of its inices. This important class of tensors generalizes the concept of symmetric matrices. As the role playe by symmetric matrices in matrix theory an quaratic optimization, symmetric tensors have a profoun role to play in tensor eigenvalue problems an polynomial optimization. A natural question can be aske about complex tensors: What is the higher orer complex tensor generalization of the Hermitian matrix? In this paper, we manage to ientify two classes of symmetric complex tensors, both of which inclue Hermitian matrices as a special case when the orer of the tensor is two. In recent years, the eigenvalue of tensor has become a topic of intensive research interest. To the best of our knowlege, a first attempt to generalize eigenvalue ecomposition of matrices can be trace back to 2000 when De Lathauwer et al. [7] introuce the so-calle higher-orer eigenvalue ecomposition. Shortly after that, Kofiis an Regalia [15] showe that blin econvolution can be formulate as a nonlinear eigenproblem. A systematic stuy of eigenvalues of tensors was pioneere by Lim [18] an Qi [21] inepenently in Various applications of tensor eigenvalues an the connections to polynomial optimization problems have been propose; cf. [22, 20, 33, 8] an the references therein. We refer the intereste reaers to the survey papers [16, 23] for more etails on the spectral theory of tensors an various applications of tensors. Computation of tensor eigenvalues is an important source for polynomial optimization [10, 17]. Essentially the problem is to maximize 2

3 or minimize a homogeneous polynomial uner the spherical constraint, which can also be use to test the (semi)-efiniteness of a symmetric tensor. This is closely relate to the nonnegativity of a polynomial function, whose history can be trace back to Hilbert [9] in 1888 where the relationship of nonnegative polynomials an sums of squares (SOS) of polynomials was first establishe. In this paper we are primarily intereste in complex polynomials/tensors that arise in the context of optimization. By nature of optimization, we are intereste in the complex polynomials that always take real values. However, it is easy to see that if no conjugate term is involve, then the only class of real-value complex polynomials is the set of real constant functions 1. Therefore, the conjugate terms are necessary for a complex polynomial to be real-value. Hermitian quaratic forms mentione earlier belong to this category, which is an active area of research in optimization; see e.g. [19, 31, 26]. In the aforementione papers [22, 20, 8] on eigenvalues of complex tensors, the associate complex polynomials however are not real-value. The aim of this paper is ifferent. We target for a systematic stuy on the nature of symmetricity for higher orer complex tensors which will lea to the property that the associate polynomials always take real values. The main contribution of the paper is to give a full characterization for the real-value conjugate complex polynomials an to ientify two classes of symmetric complex tensors, which have alreay shown potentials in the algorithms esign [2, 13, 14]. We also show that a nonnegative univariate conjugate polynomial is not necessarily an SOS polynomial, in contrast to a well-known property of its real counterpart. This paper is organize as follows. We start with the preparation of various notations an terminologies in Section 2. In particular, two types of conjugate complex polynomials are efine an their symmetric complex tensor representations are iscusse. Section 3 presents the necessary an sufficient conition for real-value conjugate complex polynomials, base on which two types of symmetric complex tensors are efine, corresponing to the two types of real-value conjugate complex polynomials. Section 4 iscusses the nonnegative properties of certain conjugate polynomials, in particular the relationship between nonnegativity an SOS for univariate complex conjugate polynomials. As an important result in this paper, we then present the efinitions an properties of eigenvalues an eigenvectors for two types of symmetric complex tensors in Section 5. Finally in Section 6, we iscuss Banach s theorem, which states that the largest absolute value of eigenvalue of a symmetric real tensor is equal to its largest singular value, an exten it to the two new types of symmetric complex tensors. 2 Preparation Throughout this paper we shall use bolface lowercase letters, capital letters, an calligraphic letters to enote vectors, matrices, an tensors, respectively. For example, a vector x, a matrix A, an a tensor F. We use subscripts to enote their components, e.g. x i being the i-th entry of a vector x, A ij being the (i, j)-th entry of a matrix A, an F ijk being the (i, j, k)-th entry of a thir orer tensor F. As usual, the fiel of real numbers an the fiel of complex numbers are enote by R an C, respectively. For any complex number z = a + ib C with a, b R, its real part an imaginary part are enote by Re z := a an Im z := b, respectively. Its moulus is enote by z := zz = a 2 + b 2, where z := a ib enotes the conjugate of z. For any vector x C n, we enote x H := x T to be the transpose of its conjugate, similar operation applying to matrices. A multivariate complex polynomial f(x) is a polynomial function of variable x C n whose 1 This shoul be ifferentiate from the notion of real-symmetric complex polynomial, sometimes also calle realvalue complex polynomial in abstract algebra, i.e., f(x) = f(x). 3

4 coefficients are complex, e.g. f(x 1, x 2 ) = x 1 + (1 i)x 2 2. A multivariate conjugate complex polynomial (sometimes abbreviate by conjugate polynomial in this paper) f C (x) is a polynomial function of variables x, x C n, which is ifferentiate by the subscript C, staning for conjugate, e.g. f C (x 1, x 2 ) = x 1 + x 2 + x 1 x 2 + (1 i)x 2 2. In particular, a general n-imensional -th egree conjugate complex polynomial can be explicitly written as summation of monomials f C (x) := l l=0 1 i 1 i k n,1 j 1 j l k n a i1...i k,j 1...j l k x i1... x ik x j1... x jl k. In this efinition, it is obvious that complex polynomials are a subclass of conjugate complex polynomials. Remark that a pure complex polynomial can never only take real values unless it is a constant. This observation follows trivially from the basic theorem of algebra. Given a -th orer complex tensor F C n 1 n, its associate multilinear form is efine as F(x 1,..., x ) := F i1...i x 1 i 1... x i, 1 i 1 n 1,...,1 i n where x k C n k for k = 1,...,. A tensor F C n 1 n is calle symmetric if n 1 = = n (= n) an every component F i1...i is invariant uner all permutations of the inices {i 1,..., i }. Closely relate to a symmetric tensor F C n is a general -th egree complex homogeneous polynomial function f(x) (or complex form) of variable x C n, i.e., f(x) := F(x,..., x) = F i1...i x i1... x i. (1) 1 i 1,...,i n In fact, symmetric tensors (either in the real omain or in the complex omain) are bijectively relate to homogeneous polynomials; see [16]. In particular, for any n-imensional -th egree complex form f(x) = a i1...i x i1... x i, 1 i 1 i n there is a uniquely efine n-imensional -th orer symmetric complex tensor F C n with F i1...i = a i1...i Π(i 1... i ) 1 i 1 i n satisfying (1), where Π(i 1... i ) is the set of all istinct permutations of the inices {i 1,..., i }. On the other han, in light of formula (1), a complex form f(x) is easily obtaine from the symmetric multilinear form F(x 1,..., x ) by letting x 1 = = x = x. 2.1 Symmetric conjugate forms an their tensor representations To iscuss higher orer conjugate complex forms an complex tensors, let us start with the well establishe properties of the Hermitian matrices. Let A C n2 with A H = A, which is not symmetric in the usual sense because A T A in general. The following conjugate quaratic form x H Ax = A ij x i x j 1 i,j n always takes real values for any x C n. In particular, we notice that each monomial in the above form is the prouct of one conjugate variable x i an one usual (non-conjugate) variable x j. 4

5 To exten the above form to higher egrees, let us consier the following special class of conjugate polynomials, to be calle symmetric conjugate forms: f S (x) := a i1...i,j 1...j x i1... x i x j1... x j. (2) 1 i 1 i n,1 j 1 j n Essentially, f S (x) is the summation of all the possible 2-th egree monomials that consist of exact conjugate variables an usual variables. Here the subscript S stans for symmetric. The following example is a special case of (2). Example 2.1 Given a -th egree complex form g(x) = 1 i 1 i n c i 1...i x i1... x i, the function g(x) 2 = c i1...i x i1... x i c j1...j x j1... x j = 1 i 1 i n 1 i 1 i n,1 j 1 j n is a 2-th egree symmetric conjugate form. 1 j 1 j n (c i1...i c j1...j ) x i1... x i x j1... x j Notice that g(x) 2 is actually a real-value conjugate polynomial. Later in Section 3 we shall show that a symmetric conjugate form f S (x) in (2) always takes real values if an only if the coefficients of any pair of conjugate monomials x i1... x i x j1... x j an x j1... x j x i1... x i are conjugate to each other, i.e., a i1...i,j 1...j = a j1...j,i 1...i 1 i 1 i n, 1 j 1 j n. As any complex form uniquely efines a symmetric complex tensor an vice versa, we observe a class of tensors representable for symmetric conjugate forms. A 2-th orer tensor F C n2 is calle partial-symmetric if for any 1 i 1 i n, 1 i +1 i 2 n F j1...j j +1...j 2 = F i1...i i +1...i 2 (j 1... j ) Π(i 1... i ), (j j 2 ) Π(i i 2 ). (3) We remark that the so-calle partial-symmetricity is a concept first stuie in [11, Section 2.1] in the framework of mixe polynomial forms, i.e., for any fixe first inices of the tensor, it is symmetric with respective to its last inices, an vise versa. It is clear that partial-symmetricity (3) is weaker than the usual symmetricity for tensors. Let us formally efine the bijection S (taking the first initial of symmetric conjugate forms) between symmetric conjugate forms an partial-symmetric complex tensors, as follows: S(F) = f S : Given a partial-symmetric tensor F C n2 with its associate multilinear form F(x 1,..., x 2 ), the symmetric conjugate form is efine as f S (x) = F(x,..., x, x,..., x) = F i1...i i +1...i 2 x i1... x i x i+1... x i2. 1 i 1,...,i 2 n S 1 (f S ) = F: Given a symmetric conjugate form f S (2), the components of the partialsymmetric tensor F C n2 are efine by F j1...j j +1...j 2 = a i1...i,i +1...i 2 Π(i 1... i ) Π(i i 2 ) 1 i 1 i n, 1 i +1 i 2 n, (j 1... j ) Π(i 1... i ), (j j 2 ) Π(i i 2 ). (4) 5

6 Accoring to the mappings efine above, the following result reaily follows. Lemma 2.2 The bijection S is well-efine, i.e., any n-imensional 2-th orer partial-symmetric tensor F C n2 uniquely efines an n-imensional 2-th egree symmetric conjugate form, an vice versa. 2.2 General conjugate forms an their tensor representations In (2), for each monomial the numbers of conjugate variables an the original variables are always equal. This restriction can be further remove. We call the following class of conjugate polynomials to be general conjugate forms: f G (x) = 1 i 1 i k n,1 j 1 j k n a i1...i k,j 1...j k x i1... x ik x j1... x j k. (5) Essentially, f G (x) is the summation of all the possible -th egree monomials, allowing any number of conjugate variables as well as the original variables in each monomial. Here the subscript G stans for general. Obviously f S (x) is a special case of f G (x), an f G (x) is a special case of f C (x). In Section 3 we shall show that a general conjugate form f G (x) which will always take real values for all x if an only if the coefficients of each pair of conjugate monomials are conjugate to each other. To this en, below we shall explicitly treat the conjugate variables as new variables. G(F) = f G : Given a symmetric tensor F C (2n) with its associate multilinear form F(x 1,..., x ), the general conjugate form of x C n is efine as ( ( ) ( ) ) x x f G (x) = F,...,. (6) x x G 1 (f G ) = F: Given a general conjugate form f G of x C n as (5), the components of the symmetric tensor F C (2n) are efine as follows: for any 1 j 1,..., j 2n, sort them in a nonecreasing orer as 1 i 1 i 2n an let k = max 1 j {i j n}, then F j1...j = a i 1...i k,(i k+1 n)...(i n). Π(i 1... i ) Similar as Lemma 2.2, the following is easily verifie; we leave its proof to the intereste reaers. Lemma 2.3 The bijection G is well-efine, i.e., any 2n-imensional -th orer symmetric tensor F C (2n) uniquely efines an n-imensional -th egree general conjugate form, an vice versa. To conclue this section we remark that a partial-symmetric tensor (representation for a symmetric conjugate form) is less restrictive than a symmetric tensor (representation for a general conjugate form), while a symmetric conjugate form is a special case of a general conjugate form. One shoul note that the imensions of these two tensor representations are actually ifferent. 3 Real-value conjugate forms an their tensor representations In this section, we stuy the two types of conjugate complex forms introuce in Section 2: symmetric conjugate forms an general conjugate forms. 6

7 3.1 Real-value conjugate polynomials Let us first focus on polynomials, an present the following general result. Theorem 3.1 A conjugate complex polynomial function is real-value if an only if the coefficients of any pair of its conjugate monomials are conjugate to each other, i.e., any two monomials au C (x) an bv C (x) with a an b being their coefficients satisfying u C (x) = v C (x) must follow that a = b. Specifically, applying Theorem 3.1 to the two classes of conjugate forms that we just introuce, the conitions for them to always take real values can now be characterize: Corollary 3.2 A symmetric conjugate form f S (x) = is real-value if an only if 1 i 1 i n,1 j 1 j n a i1...i,j 1...j x i1... x i x j1... x j a i1...i,j 1...j = a j1...j,i 1...i 1 i 1 i n, 1 j 1 j n. (7) A general conjugate form f G (x) = is real-value if an only if 1 i 1 i k n,1 j 1 j k n a i1...i k,j 1...j k x i1... x ik x j1... x j k a i1...i k,j 1...j k = a j1...j k,i 1...i k 1 i 1 i k n, 1 j 1 j k n, 0 k. Before getting into the technical proof of Theorem 3.1, let us first present an alternative representation of real-value symmetric conjugate forms, as a consequence of Corollary 3.2. Proposition 3.3 A symmetric conjugate form f S (x) is real-value if an only if f S (x) = k K α k g k (x) 2, where g k (x) is a complex form an α k R for all k K. Proof. The if part is trivial. Next we prove the only if part of the proposition. If f S (x) is realvalue, by Corollary 3.2 we have (7). Then for any 1 i 1 i n an 1 j 1 j n, the sum of the conjugate pair satisfies a i1...i,j 1...j x i1... x i x j1... x j + a j1...j,i 1...i x j1... x j x i1... x i = a i1...i,j 1...j x i1... x i x j1... x j + a i1...i,j 1...j x j1... x j x i1... x i = x i1... x i + a i1...i,j 1...j x j1... x j 2 x i1... x i 2 a i1...i j 1...j x j1... x j 2. Summing up all such pairs, the conclusion follows. Similarly we have the following result for general conjugate forms. 7

8 Proposition 3.4 A general conjugate form f G (x) is real-value if an only if f G (x) = k K α k g k (x) 2, where g k (x) is a complex polynomial an α k R for all k K. Let us now turn to proving Theorem 3.1. We first show the if part of the theorem, which is quite straightforwar. To see this, for any pair of conjugate monomials (incluing self-conjugate monomial as a special case) of a conjugate complex polynomial: au C (x) an bv C (x) with a, b C being their coefficients an u C (x) = v C (x), if a = b, then au C (x) + bv C (x) = au C (x) + au C (x) = au C (x) + au C (x) = au C (x) + bv C (x), implying that au C (x) + bv C (x) is real-value. Since all the conjugate monomials of a conjugate complex polynomial can be partitione by conjugate pairs an self-conjugate monomials, the result follows immeiately. To procee to the only if part of the theorem, let us consier an easier case of the univariate conjugate polynomials. Lemma 3.5 A univariate conjugate complex polynomial l=0 l b k,l kx k x l k = 0 for all x C if an only if all its coefficients are zeros, i.e., b k,l k = 0 for all 0 l an 0 k l. Proof. Let x = ρe iθ, the ientity can be rewritten as ( l ) b k,l k e i(l 2k)θ ρ l = 0 ρ (0, ), θ [0, 2π). (8) l=0 For any fixe θ, the function can be viewe as a polynomial with respect to ρ. coefficient of the highest egree monomial ρ must be zero, i.e., Therefore the b k, k e i( 2k)θ = 0 Consequently we have for any θ [0, 2π), θ [0, 2π). an Re (b k, k ) cos(( 2k)θ) Im (b k, k ) cos(( 2k)θ) + The first an secon parts of (9) can be respectively simplifie as Re (b k, k ) cos(( 2k)θ) = an Im (b k, k ) sin(( 2k)θ) = Im (b k, k ) sin(( 2k)θ) = 0 (9) Re (b k, k ) sin(( 2k)θ) = 0. (10) 1 2 Re (b k, k + b k,k ) cos(( 2k)θ) is o 2 2 Re (b k, k + b k,k ) cos(( 2k)θ) + Re (b /2,/2 ) is even Im (b k, k b k,k ) sin(( 2k)θ).

9 By the orthogonality of the trigonometric functions, it further leas to Similarly, (10) implies Re (b k, k + b k,k ) = Im (b k, k b k,k ) = 0 k = 0, 1,...,. Re (b k, k b k,k ) = Im (b k, k + b k,k ) = 0 k = 0, 1,...,. Combining the above two sets of ientities yiels b k, k = 0 k = 0, 1,...,. The egree of the function in (8) (in terms of ρ) is then reuce by 1. The esire result can thus be proven by inuction. Let us now exten Lemma 3.5 to the general multivariate conjugate polynomials. Lemma 3.6 An n-imensional -th egree conjugate complex polynomial l l=0 1 i 1 i k n,1 j 1 j l k n b i1...i k,j 1...j l k x i1... x ik x j1... x jl k = 0 for all x C n if an only if all its coefficients are zeros, i.e., b i1...i k,j 1...j l k = 0 for all 0 l, 0 k l, 1 i 1 i k n, 1 j 1 j k n. Proof. We shall prove the result by inuction on the imension n. The case n = 1 is alreay shown in Lemma 3.5. Suppose the claim hols for all positive integers no more than n 1. Then for the imension n, the conjugate polynomial f C (x) can be rewritten accoring to the egrees of x 1 an x 1 as f C (x) = l l=0 x 1 k x 1 l k g lk C (x 2,..., x n ). For any given x 2,..., x n C, taking f C as a univariate conjugate polynomial of x 1, by Lemma 3.5 we have g lk C (x 2,..., x n ) = 0 0 l, 0 k l. For any given (l, k), as gc lk(x 2,..., x n ) is a conjugate polynomial of imension at most n 1, by the inuction hypothesis all the coefficients of gc lk are zeros. Observing that all the coefficients of f C are istribute in the coefficients of gc lk for all (l, k), the claime result is proven for the imension n. The proof is thus complete by inuction. With Lemma 3.6 at han, we can finally complete the only if part of Theorem 3.1. Suppose a conjugate polynomial f(x) is real-value for all x C n. Clearly we have f(x) f(x) = 0 for all x C n, i.e., l ( bi1...i k,j 1...j l k b j1...j l k,i 1...i k ) xi1... x ik x j1... x jl k. l=0 1 i 1 i k n,1 j 1 j l k n By Lemma 3.6 it follows that b i1...i k,j 1...j l k b j1...j l k,i 1...i k = 0 for all 0 l, 0 k l, 1 i 1 i k n, 1 j 1 j k n, proving the only if part of Theorem 3.1. With Theorem 3.1, in particular Corollary 3.2, we are in a position to characterize the tensor representations for real-value conjugate forms. 9

10 3.2 Conjugate partial-symmetric tensors As any symmetric conjugate form uniquely efines a partial-symmetric tensor (Lemma 2.2), it is interesting to see more structure tensor representations for real-value symmetric conjugate forms. Definition 3.7 A 2-th orer tensor F C n2 is calle conjugate partial-symmetric if (1) F i1...i i +1...i 2 = F j1...j j +1...j 2 (j 1... j ) Π(i 1... i ), (j j 2 ) Π(i i 2 ), an (2) F i1...i i +1...i 2 = F i+1...i 2 i 1...i hol for any 1 i 1 i n, 1 i +1 i 2 n. We remark that when = 1, a conjugate partial-symmetric tensor is simply a Hermitian matrix. The conjugate partial-symmetric tensors an the real-value symmetric conjugate forms are connecte as follows. Lemma 3.8 Any n-imensional 2-th orer conjugate partial-symmetric tensor F C n2 uniquely efines (uner S) an n-imensional 2-th egree real-value symmetric conjugate form, an vice versa (uner S 1 ). Proof. For any conjugate partial-symmetric tensor F, f S = S(F) satisfies f S (x) = F(x,..., x, x,..., x) = = = 1 i 1,...,i 2 n 1 i 1,...,i 2 n 1 i 1,...,i 2 n = f S (x), F i1...i i +1...i 2 x i1... x i x i+1... x i2 F i1...i i +1...i 2 x i1... x i x i+1... x i2 F i+1...i 2 i 1...i x i+1... x i2 x i1... x i implying that f S is real-value. On the other han, for any real-value symmetric conjugate form f S (x) in (2), it follows from Corollary 3.2 that a i1...i,j 1...j = a j1...j,i 1...i for all possible (i 1,..., i, j 1,..., j ). By (4), its tensor representation F = S 1 (f S ) with F i1...i i +1...i 2 = a i1...i,i +1...i 2 Π(i 1... i ) Π(i i 2 ) satisfies the 2n conition in Definition 3.7, proving the conjugate partial-symmetricity of F. Below is a useful property of the conjugate partial-symmetric tensors. Lemma 3.9 For any conjugate partial-symmetric tensor F C n2 an vectors x 1,..., x 1 C n, f C (z) := F(z, x 1,..., x 1, z, x 1,..., x 1 ) is a Hermitian quaratic form of the variable z C n, i.e., the matrix Q := F(, x 1,..., x 1,, x 1,..., x 1 ) is a Hermitian matrix. 10

11 Proof. To prove Q ij = Q ji for all 1 i, j n we only nee to show F(e i, x 1,..., x 1, e j, x 1,..., x 1 ) = F(e j, x 1,..., x 1, e i, x 1,..., x 1 ), where e i enotes the vector whose i-th component is 1 an others are zeros for i = 1,..., n. In fact, accoring to Definition 3.7 F(e i, x 1,..., x 1, e j, x 1,..., x 1 ) = = = 1 i 1,...,i 1,j 1,...,j 1 n 1 j 1,...,j 1,i 1,...,i 1 n 1 j 1,...,j 1,i 1,...,i 1 n = F(e j, x 1,..., x 1, e i, x 1,..., x 1 ). F ii1...i 1 jj 1...j 1 x 1 i 1... x 1 i 1 x 1 j 1... x 1 j 1 F jj1...j 1 ii 1...i 1 x 1 j 1... x 1 j 1 x 1 i 1... x 1 i 1 F jj1...j 1 ii 1...i 1 x 1 j 1... x 1 j 1 x 1 i 1... x 1 i 1 In general, it can be shown for any conjugate partial-symmetric tensor F C n2 x 1,..., x t C n with 1 t < that an vectors F(,...,, x 1,..., x t,,...,, x 1,..., x t ) t t is a conjugate partial-symmetric tensor in C n2 2t. 3.3 Conjugate super-symmetric tensors Similar as for the real-value symmetric conjugate forms, we have the following tensor representations for the real-value general conjugate forms. Definition 3.10 A 2n-imensional tensor F C (2n) is calle conjugate super-symmetric if (1) F is symmetric, i.e., F i1...i = F j1...j (j 1... j ) Π(i 1... i ), an (2) F j1...j = F i1...i if i k j k = n hols for all 1 k hol for any 1 i 1 i n. Clearly, the conjugate super-symmetricity is stronger than the orinary symmetricity for complex tensors. Uner the mapping G efine in Section 2.2, we have the following tensor representations for the real-value general conjugate forms. Proposition 3.11 Any 2n-imensional -th orer conjugate super-symmetric tensor F C (2n) uniquely efines (uner G) an n-imensional -th egree real-value general conjugate form, an vice versa (uner G 1 ). 4 Nonnegative real-value conjugate polynomials An important aspect of polynomials is the theory of nonnegativity. However, most existing results only apply for polynomials in real variables, for the reason that such polynomials are real-value. Since we have introuce several classes of complex polynomials which are real-value, the question 11

12 about their nonnegativity naturally arises. In particular, in this section, we stuy the relationship between nonnegativity an sums of squares (SOS) for univariate conjugate polynomials. In the real omain, this problem was completely solve by Hilbert [9] in 1888, where the only three general classes of real polynomials whose nonnegativity is equivalent to SOS are: (1) univariate polynomials, (2) quaratic polynomials, an (3) bivariate quartic polynomials. However, relationship between nonnegative complex polynomials an SOS has not been establishe explicitly in the literature as far as we know. This section aims to fill in this gap, using the notion of conjugate polynomials. To begin with, let us start with the efinitions of nonnegativity an SOS for a conjugate complex polynomial. Definition 4.1 A conjugate complex polynomial f C (x) is nonnegative if Re f C (x) 0 x C n. In particular, a real-value conjugate polynomial f C (x) is nonnegative if f C (x) 0 for all x C n. Definition 4.2 A conjugate complex polynomial f C (x) is SOS if there exist conjugate complex polynomials gc 1,..., gm C such that Re f C (x) = m gc(x) i 2. i=1 In particular, a real-value conjugate polynomial f C (x) is SOS if f C (x) = m i=1 gi C (x) 2. Our next proposition states that it is sufficient to focus on the real-value conjugate polynomials as long as the nonnegativity is concerne. Proposition 4.3 Any conjugate complex polynomial f C (x) can be uniquely written as g C (x) + ih C (x) where both g C an h C are real-value conjugate polynomials. Proof. As any conjugate complex polynomial can be partitione by pairs of conjugate monomials an self-conjugate monomials, it suffices to rewrite the summation of a pair of conjugate monomials (a self-conjugate monomial can be split into two halves an then taken as a pair). Let au C (x) an bv C (x) be a pair of conjugate monomials of f C with a, b C being their coefficients an u C (x) = v C (x). Denote an so (α, β) = ( a + b 2, a b ), 2i au C (x) + bv C (x) = (αu C (x) + αv C (x)) + i ( βu C (x) + βv C (x) ), where αu C (x) + αv C (x) an βu C (x) + βv C (x) are both real-value by Theorem 3.1. Therefore the ecomposition of f C as real an complex parts is constructe. The uniqueness of the ecomposition is obvious. Let us now focus on univariate conjugate polynomials, i.e., h C (x) = l=0 l a k,l k x k x l k. (11) 12

13 It is obvious that SOS implies nonnegativity, but the other way roun implication is the topic of stuy in this section. Uner certain circumstance, the equivalence between SOS an nonnegativity can be achieve, such as Hilbert s classical results in the real omain as we mentione earlier. In particular, in the real omain a univariate polynomial is nonnegative if an only if it is SOS representable, while this equivalence oes not hol anymore for bivariate polynomials an beyon. Another example is the so-calle Riesz-Féjer theorem (see e.g. [34]): Theorem 4.4 (Riesz-Féjer) A univariate complex polynomial Re h(x) 0 for all x = 1 if an only if there exist c 0, c 1,..., c C such that 2 Re h(x) = c k x k. This states a special class of univariate complex polynomials (also a special class of univariate conjugate polynomials) whose nonnegativity is equivalent to SOS if the variable x C lies on the unit circle of the complex plane. Although the real part of any univariate complex polynomial of x can be viewe as a special real polynomial with two variables (Re x, Im x), the relationship between nonnegativity an SOS remains unclear uner this light. To further stuy this problem for the univariate conjugate polynomial (11), let us first generalize Theorem 4.4. Proposition 4.5 A univariate conjugate polynomial Re h C (x) 0 for all x = 1 if an only if there exist c 0, c 1,..., c C such that 2 Re h C (x) = c k x k. Proof. The if part is trivial. Let us prove the only if part. Since x = 1 we have x = x 1 an then x k x l k = x l 2k. Consequently, ( ) ( l ) ( Re h C (x) = Re a k,l k x l 2k = Re b l x l = Re (b l + b l )x l) + Re b 0, where l=0 b l = l 2 +l l= a k,l+k l 0, 2 a k l,k l < 0. Let us efine g(x) = l=1 (b l + b l )x l + b 0, which is a univariate complex polynomial. If Re g(x) = Re h C (x) 0 for all x = 1, by applying Riesz-Féjer theorem (Theorem 4.4) on g(x), there exist c 0, c 1,..., c C such that 2 Re h C (x) = Re g(x) = c k x k, proving the only if part. However if we o rop the constraint x = 1 in Proposition 4.5, the equivalence oes not hol for univariate conjugate polynomials in general. Theorem 4.6 For a univariate conjugate polynomial h C (x) efine by (11), Re h C (x) 0 for all x C oes not imply that it is SOS representable. l=1 13

14 Proof. Suppose on the contrary, for any nonnegative univariate conjugate polynomial h C (x), there exist univariate conjugate polynomials gc 1,..., gm C such that Re h C (x) = m gc(x) i 2. Let us consier a general nonnegative bivariate real polynomial p(y, z) = l=0 i=1 l p k,l k y k z l k 0 y, z R. Denote x = (y + iz)/2, an so y = x + x an z = i(x x). Consequently p(y, z) can be written as p(y, z) = l=0 l p k,l k (x + x) k ( ix + ix) l k := h C (x). It is obvious that h C (x) is a real-value nonnegative univariate conjugate polynomial. By the assumption, there exist univariate conjugate polynomials gc 1,..., gm C such that p(y, z) = h C (x) = Re h C (x) = m gc(x) i 2 = i=1 m ( (Re g i C (x)) 2 + (Im gc(x) i 2 ) ). Notice that both Re gc i (x) an Im gi C (x) can be rewritten as real polynomials of the real variables (y, z) by replacing x with (y + iz)/2. Therefore we get an SOS representation of any nonnegative bivariate real polynomial p(y, z), which contraicts the fact that nonnegativity of a bivariate real polynomial is not necessarily SOS. To conclue this section, we present the following result extene from the real case, which states that any real convex form is nonnegative (see e.g. [25]). Proposition 4.7 Any real-value convex general conjugate form f G (x) (incluing symmetric conjugate form f S (x) as a special case) is nonnegative. Proof. Suppose real-value f G (x) is convex at any x C n. Define ˆf x,y : R R with ˆf x,y (t) = f G (x + ty). It is well know in convex analysis that ˆf x,y (t) is a convex function of t R for all x, y C n. Let the tensor representation of f G, G 1 (f G ) = F, an by (6) we have ( ( ) ( ) ) x + ty x + ty ˆf x,y (t) = F,...,. x + ty x + ty As F is symmetric, irect computation shows that (( ) ( ) ( ) ) y x + ty x + ty ˆf x,y(t) = F,,...,, y x + ty x + ty 1 an furthermore ˆf x,y(t) = ( 1)F (( ) y, y ( ) y, y 14 i=1 ( ) ( ) ) x + ty x + ty,..., 0 x + ty x + ty 2

15 for all t R an x, y C n. In particular, by letting t = 0 an y = x we get ( ( ) ( ) ) x x ˆf x,x(0) = ( 1)F,..., = ( 1) f G (x) 0 x x for all x C n, proving the nonnegativity of f G (x). 5 Eigenvalues an eigenvectors of complex tensors As mentione earlier, Lim [18] an Qi [21] inepenently propose to systematically stuy the eigenvalues an eigenvectors for real tensors, though in their prior works De Lathauwer et al. [7] an Kofiis an Regalia [15] iscusse various aspects of tensor eigenproblems alreay. Subsequently, the topic has attracte much attention ue to the potential applications in magnetic resonance imaging, polynomial optimization theory, quantum physics, statistical ata analysis, higher orer Markov chains, an so on. After that, this stuy was also extene to complex tensors [22, 20, 8] without consiering the conjugate variables. Zhang an Qi in [33] propose the so-calle Q- eigenvalues of complex tensors: Definition 5.1 (Zhang an Qi [33]) A scalar λ is calle a Q-eigenvalue of a symmetric complex tensor H, if there exists a vector x calle Q-eigenvector, such that H(, x,..., x) = λx 1 x H x = 1 λ R. However, as the corresponing complex tensor oes not have conjugate-type symmetricity, the eigenvalue efine above oes not specialize to the classical eigenvalues of Hermitian matrices. In particular, λ R is put in the system (12). Now with all the new notions introuce in the previous sections in particular the bijection between conjugate partial-symmetric tensors an real-value symmetric conjugate forms, an the bijection between conjugate super-symmetric tensors an realvalue general conjugate forms we are able to present new efinitions an properties of eigenvalues for complex tensors, which are more naturally relate to that of the Hermitian matrices. 5.1 Definitions an properties of eigenvalues Let us first introuce two types of eigenvalues for conjugate partial-symmetric tensors an conjugate super-symmetric tensors. Definition 5.2 λ C is calle C-eigenvalue of a conjugate partial-symmetric tensor F, if there exists a vector x C n calle C-eigenvector, such that F(, x,..., x, x,..., x) = λx 1 x H x = 1. (12) (13) 15

16 Definition 5.3 λ C is calle G-eigenvalue of a conjugate super-symmetric tensor F, if there exists a vector x C n calle G-eigenvector, such that (( ) ( ) ( ) ) ( ) x x x F,,..., = λ x x x (14) 1 x H x = 1. In fact, these two types of eigenvalues efine above are always real, although they are efine in the complex omain. This property generalizes the well-known property of the Hermitian matrices. In particular, Definition 5.2 inclues eigenvalues of Hermitian matrices as a special case when = 1. Proposition 5.4 Any C-eigenvalue of a conjugate partial-symmetric tensor is always real; so is any G-eigenvalue of a conjugate super-symmetric tensor. Proof. Suppose (λ, x) is a C-eigenvalue an C-eigenvector pair of a conjugate partial-symmetric tensor F. Multiplying x on both sies of the first equation in (13), we get F(x,..., x, x,..., x) = λx T x = λ. As F is conjugate partial-symmetric, the symmetric conjugate form F(x,..., x, x,..., x) is realvalue, an so is λ. Next, suppose (λ, x) is a G-eigenvalue an G-eigenvector pair of a conjugate super-symmetric tensor F. Multiplying ( x x) on both sies of the first equation in (14) yiels ( ( ) ( ) ) ( ) x x x T ( ) x F,..., = λ = 2λx H x = 2λ. x x x x ( ( ) ( ) ) x x As F is conjugate super-symmetric, the general conjugate form F,..., is real-value, x x an so is λ. As a consequence of Proposition 5.4, one can efine the C-eigenvalue λ R an its corresponing C-eigenvector x C n for a conjugate partial-symmetric tensor F equivalently as follows. Proposition 5.5 λ C is a C-eigenvalue of a conjugate partial-symmetric tensor F, if an only if there exists a vector x C n, such that F(x,..., x, x,..., x, ) = λx 1 x H x = 1. Proof. Suppose a C-eigenvalue λ an C-eigenvector x satisfy (13). By Proposition 5.4 we know that λ R. Therefore λx = λx = F(, x,..., x, x,..., x) = F(, x,..., x, x,..., x) = F(x,..., x, x,..., x, ), (15)

17 where the last equality is ue to the conjugate partial-symmetricity of F. Finally, the converse can be proven similarly. One important property of the Z-eigenvalues for real symmetric tensors is that they can be fully characterize by the KKT solutions of a certain optimization problem [18, 21]. At a first glance, this property may not hol for C-eigenvalues an G-eigenvalues since the real-value complex functions are not analytic. Therefore, irect extension of the KKT conition of an optimization problem with such objective function may not be vali. However, this class of functions is inee analytic if we treat the complex variables an their conjugates as a whole ue to the so-calle Wirtinger calculus [24] in German literature, evelope in the early 20th century. In the optimization context, without noticing the Wirtinger calculus, Branwoo [4] first propose the notion of complex graient. In particular, the graient of a real-value complex function can be taken as ( x, x). Intereste reaers are referre to [27] for more iscussions on the Wirtinger calculus in optimization with complex variables. With the help of Wirtinger calculus, we are able to characterize C-eigenvalues an C-eigenvectors in terms of the KKT solutions. Therefore many optimization techniques can be applie to fin the C-eigenvalues/eigenvectors for a conjugate partial-symmetric tensor. Proposition 5.6 x C n is a C-eigenvector associate with a C-eigenvalue λ R for a conjugate partial-symmetric tensor F if an only if x is a KKT point of the optimization problem max x H x=1 F(x,..., x, x,..., x) with Lagrange multiplier being λ an the corresponing objective value being λ. Proof. Denote µ to be the Lagrange multiplier associate with the constraint x H x = 1. The KKT conition gives rise to the equations F(, x,..., x, x,..., x) µx = 0 1 F(x,..., x,, x,..., x) µx = 0 1 x H x = 1. The conclusion follows immeiately by comparing the above with (13) an (15). Similarly, we have the following characterization. Proposition 5.7 x C n is a G-eigenvector associate with a G-eigenvalue λ R for a conjugate super-symmetric tensor F if an only if x is a KKT point of the optimization problem ( ( ) ( ) ) x x max F,..., x H x=1 x x with Lagrange multiplier being λ an the corresponing objective value being λ. 17

18 5.2 Eigenvalues of complex tensors an their relations Although the efinitions of the C-eigenvalue, the G-eigenvalue, an the previously efine Q- eigenvalue involve ifferent tensor spaces, they are inee closely relate. Our main result in this section essentially states that the Q-eigenvalue is a special case of the C-eigenvalue, an the C-eigenvalue is a special case of the G-eigenvalue. Theorem 5.8 Denote H C n to be a complex tensor an efine F = H H C n2. It hols that (i) H is symmetric if an only if F is conjugate partial-symmetric; (ii) All the C-eigenvalues of F are nonnegative; (iii) λ 2 is a C-eigenvalue of F if an only if λ is a Q-eigenvalue of H. Proof. (i) This equivalence can be easily verifie by the efinition of conjugate partial-symmetricity (Definition 3.7). (ii) Let x C n be a C-eigenvector associate with a C-eigenvalue λ R of F. By multiplying x on both sies of the first equation in (13), we obtain λ = F(x,..., x, x,..., x) = (H H)(x,..., x, x,..., x) = H(x,..., x) H(x,..., x) = H(x,..., x) H(x,..., x) = H(x,..., x) 2 0. (iii) Suppose x C n is a Q-eigenvector associate with a Q-eigenvalue λ R of H. By the efinition (12) we have x H x = 1 an H(, x,..., x) = λx, an so 1 H(x,..., x) = λx T x = λ. By the similar erivation in the proof of (ii), we get F(, x,..., x, x,..., x) = H(, x,..., x) H(x,..., x) = H(, x,..., x) λ = λx λ = λ 2 x, implying that λ 2 is a C-eigenvalue of F. On the other han, suppose x C n is a C-eigenvector associate with a nonnegative C- eigenvalue λ 2 of F. Then by (15) we have x H x = 1 an H(x,..., x) H(, x,..., x) = H(x,..., x) H(x,..., x, ) = F(x,..., x, x,..., x, ) = λ 2 x, (16)

19 where the first equality is ue to the symmetricity of H. This leas to H(x,..., x) 2 = λ 2. Let H(x,..., x) = λe iθ with θ [0, 2π), an further efine y = xe iθ/. We then get H(y,..., y) = H(xe iθ/,..., xe iθ/ ) = (e iθ/ ) H(x,..., x) = e iθ λe iθ = λ. Now we are able to verify that y is a Q-eigenvector associate with Q-eigenvalue λ of H. First y H y = (xe iθ/ ) H xe iθ/ = 1, an secon by (16) λ 2 x = H(x,..., x) H(, x,..., x) = λe iθ H(, ye iθ/,..., ye iθ/ ) = λe iθ (e iθ/ ) 1 H(, y,..., y), we finally get H(, y,..., y) = λxe iθ/ = λye iθ/ e iθ/ = λy. 1 In Section 2, by efinition a symmetric conjugate form is a special general conjugate form. Hence in terms of their tensor representations, a conjugate partial-symmetric tensor is a special case of conjugate super-symmetric tensor, although they live in ifferent tensor spaces. To stuy the relationship between the C-eigenvalues an the G-eigenvalues, let us introuce an embee conjugate partial-symmetric tensor F C n2 to the space of C (2n)2. The conjugate super-symmetric tensor G C (2n)2 corresponing to F is then efine by G j1...j 2 = { Fi1...i 2 /( 2 ) (j1... j 2 ) Π(i 1,..., i, i +1 + n,..., i 2 + n) 0 otherwise. (17) For example when = 1, a conjugate partial-symmetric tensor ( is simply a Hermitian ) matrix O A/2 A C n2. Then its embee conjugate super-symmetric tensor is A T C /2 O (2n)2, an clearly we have ( ) x T ( ) ( ) x T O A/2 x Ax = x A T. /2 O x ( ( ) ( ) ) x x In general it is straightforwar to verify that F(x,..., x, x,..., x) = G,...,. Base x x on this, we are le to the following relationship between the C-eigenvalues an the G-eigenvalues, whose proof is similar to that of Theorem 5.8. Theorem 5.9 If G C (2n)2 is a conjugate super-symmetric tensor inuce by a conjugate partialsymmetric tensor F C n2 accoring to (17), then λ is a C-eigenvalue of F if an only if λ/2 is a G-eigenvalue of G. 2 19

20 6 Extening Banach s theorem to the real-value conjugate forms A classical result originally ue to Banach [3] states that if L(x 1,..., x ) is a continuous symmetric -linear form, then sup{ L(x 1,..., x ) x 1 1,..., x 1} = sup{ L(x,..., x) x 1}. (18) In the real tensor setting, where x R n an L is a symmetric multilinear form associate with a real symmetric tensor L R n, (18) states that the largest singular value [18] of L is equal to the largest absolute value of eigenvalue [21] of L, i.e., max L(x,..., x ) = max x T x=1, x R n (x i ) T x i =1, x i R n, i=1,..., L(x 1,..., x ). (19) Alternatively, (19) is essentially equivalent to the fact that the best rank-one approximation of a real symmetric tensor can be obtaine at a symmetric rank-one tensor [5, 32]. A recent evelopment on this topic for special classes of real symmetric tensors can be foun in [6]. In this section, we shall exten the Banach s theorem to the symmetric conjugate forms (the conjugate partial-symmetric tensors) an the general conjugate forms (the conjugate super-symmetric tensors). 6.1 Equivalence for conjugate super-symmetric tensors Let us start with the conjugate super-symmetric tensors, which are a generalization of conjugate partial-symmetric tensors. A key result le to the equivalence (Theorem 6.2) is the following. Lemma 6.1 For a given real tensor F R n, if F(x 1,..., x ) = F(x π(1),..., x π() ) for any x 1,..., x R n an any permutation π of {1,..., }, then F is symmetric. Proof. Denote e i to be the vector whose i-th component is 1 an others are zeros for i = 1,..., n. For any permutation π of {i 1,..., i }, we have F π(i1 )...π(i ) = F(e π(i 1),..., e π(i ) ) = F(e i 1,..., e i ) = F i1...i, an the conclusion follows. Our first result in this section extens (19) to any conjugate super-symmetric tensors in the complex omain. Theorem 6.2 For any conjugate super-symmetric tensor G C (2n), we have max ( ( ) ( ) ) (( ) ( )) x x x G 1 x,..., = max Re G x x (x ) H x i =1, i=1,..., x 1,..., x. (20) x H x=1 ( ) (( ) ( )) Re x Proof. Let y i i x = Im x i R 2n 1 x for i = 1,...,. We observe that Re G x 1,..., x is also a multilinear form with respect to y 1,..., y. As a result, we are able to fin a real tensor F R (2n) such that F ( y 1,..., y ) (( ) ( )) x 1 x = Re G,...,. (21) 20 x 1 x

21 As G is conjugate super-symmetric, for any y 1,..., y R 2n an any permutation π of {1,..., }, one has F ( y 1,..., y ) (( ) ( )) (( ) ( )) x 1 x x π(1) x π() = Re G,..., = Re G,..., = F ( y π(1),..., y π()). x 1 x By Lemma 6.1 we have that the real tensor F is symmetric. Finally, noticing that (y i ) T y i = (x i ) H x i for i = 1,...,, the conclusion follows immeiately by applying (19) to F an then using the equality (21). x π(1) x π() 6.2 Equivalence for conjugate partial-symmetric tensors For a conjugate partial-symmetric tensor F C n2, as it is a special case of conjugate supersymmetric tensors, one can certainly embe F into a super-symmetric structure G C (2n)2 using (17). By applying Theorem 6.2 to G an rewrite its associate form in terms of F we get an equivalent expression as (20). However, this expression is not succinct. Taking the case = 2 (egree 4) for example, this woul lea to max F(x, x, x, x) = max f(x 1, x 2, x 3, x 4 ), x H x=1 (x i ) H x i =1, i=1,2,3,4 where f(x 1, x 2, x 3, x 4 ) := 1 ( F(x 6 1, x 2, x 3, x 4 ) + F(x 1, x 3, x 2, x 4 ) + F(x 1, x 4, x 2, x 3 ) ) +F(x 2, x 3, x 1, x 4 ) + F(x 2, x 4, x 1, x 3 ) + F(x 3, x 4, x 1, x 2 ). Instea, one woul hope to get max x H x=1 F(x,..., x, x,..., x) = max (x i ) H x i =1, i=1,...,2 However, this oes not hol in general. The main reason is that (( ) ( )) x 1 x 2 G,..., F(x 1,..., x, x +1,..., x 2 ), x 1 x 2 Re F(x 1,..., x, x +1,..., x 2 ). (22) which is easily observe since its left han sie is invariant uner the permutation of (x 1,..., x 2 ) while its right han sie is not. In particular, (22) only hols for = 1, i.e, Hermitian matrices; see the following result an Example 6.4. Proposition 6.3 For any Hermitian matrix Q C n n, it hols that (L) max z H Qz = max Re x T Qy. z H z=1 x H x=y H y=1 (R) Furthermore, for any optimal solution (x, y ) of (R) with x + y 0, (x + y )/ x + y is an optimal solution of (L) as well. 21

22 Proof. Denote v(l) an v(r) to be the optimal values of (L) an (R), respectively. Noticing that Re x T Qy = 1 2 (xt Qy + x T Qy), by the optimality conition of (R) we have that Qy 2λx = 0 Qy 2λx = 0 Qx 2µy = 0 Qx 2µy = 0 (x ) H x = 1 (y ) H y = 1, where λ an µ are the lagrangian multipliers of the constraints x H x = 1 an y H y = 1, respectively. The summation of the first two equations in (23) leas to 2Re (x ) T Qy = (x ) T Qy + (x ) T Qy = 2λ(x ) T x + 2λ(x ) T x = 4λ(x ) H x = 4λ. Similarly, the summation of the thir an fourth equations in (23) leas to which further leas to 2Re (x ) T Qy = 4µ, (23) v(r) = Re (x ) T Qy = 2λ = 2µ. (24) Moreover, the summation of the first an thir equations in (23) yiels which further leas to Q(y + x ) 2λ(x + y ) = 0, (y + x ) H Q(y + x ) = 2λ(y + x ) H (x + y ) = 2λ x + y 2. Let z = (x + y )/ x + y. Clearly z is a feasible solution of (L). By (24) we have that (z ) H Qz = 2λ = Re (x ) T Qy = v(r). This implies that v(l) v(r). Notice that (R) is a relaxation of (L) an hence v(l) v(r). Therefore we conclue that v(r) = v(l), an an optimal solution z of (L) is constructe from an optimal solution (x, y ) of (R). Example 6.4 Let F C 24 with F 1122 = F 2211 = 1 an other entries being zeros. Clearly F is conjugate partial-symmetric. We have (22) fail to hol since: F(x, x, x, x) = x 1 2 x x 2 2 x x 1 2 x ( x x 2 2 ) 2 = 1 2 for any x C2 with x H x = 1. F(x, y, z, w) = x 1 y 1 z 2 w 2 + x 2 y 2 z 1 w 1 = 1 for x = y = (1, 0) T an z = w = (0, 1) T. Although (22) oes not hol, we have a relaxe version of the general equivalence result for the conjugate partial-symmetric tensors. By Lemma 3.9, F(x 1,..., x, x 1,..., x ) always takes real values, an we have the following result. Theorem 6.5 For any conjugate partial-symmetric tensor F C n2, we have max x H x=1 F(x,..., x, x,..., x) = max (x i ) H x i =1, i=1,..., F(x 1,..., x, x 1,..., x ) 22