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It was conjectured in the paper “Stationary probability vectors of higher-order Markov chains” (Li and Zhang, 2015 [7] ) that if the set of stationary vectors of the second-order Markov chain contains k -interior points of the ( k − 1 ) -dimensional face of the simplex Ω n then every vector in the ( k − 1 ) -dimensional face is a stationary vector. In this paper, we provide counterexamples to this conjecture. [ABSTRACT FROM AUTHOR]

Abstract: In an earlier paper (R. Bhatia, T. Jain, Higher order derivatives and perturbation bounds for determinants, Linear Algebra Appl. 431 (2009) 2102–2108) we gave formulas for derivatives of all orders for the map that takes a matrix to its determinant. In this paper we continue that work, and find expressions for the derivatives of all orders for the antisymmetric tensor powers and for the coefficients of the characteristic polynomial. We then evaluate norms of these derivatives, and use them to obtain perturbation bounds. [Copyright &y& Elsevier]

A positive semidefinite matrix can be written as a Gramian matrix, and a completely positive matrix can therefore be written as a Gramian matrix of some nonnegative vectors. In this paper, we introduce Gramian tensors and study 2-dimension completely positive tensors and { 0 , 1 } − C P tensors. Also investigated are the complete positive multi-hypergraph which are a generalized form of cp graphs. We also provide a necessary and sufficient condition for a 2-dimensional tensor to be completely positive. [ABSTRACT FROM AUTHOR]

The operator Schmidt rank of an operator acting on the tensor product C n ⊗ C m is the number of terms in a decomposition of the operator as a sum of simple tensors with factors forming orthogonal families in their respective matrix algebras. It has been known that for unitary operators acting on two copies of C 2 , the operator Schmidt rank can only take the values 1, 2, and 4, the value 3 being forbidden. In this paper, we settle an open question, showing that the above obstruction is the only one occurring. We do so by constructing explicit examples of bipartite unitary operators of all possible operator Schmidt ranks, for arbitrary dimensions ( n , m ) ≠ ( 2 , 2 ) . [ABSTRACT FROM AUTHOR]

The geometric measure of entanglement of a multipartite pure state is defined it terms of its geometric distance from the set of separable pure states. The quantum eigenvalue problem is derived to compute the separable pure state nearest to the given multipartite pure state. Computing the modulus largest quantum eigenvalue for a multipartite pure state is equivalent to finding the best complex rank-one approximation of the complex unit tensors, associated with the multipartite pure states. This paper is devoted to present a complex-valued neural networks approach for the computation of the quantum eigenvalue problem for multipartite pure states. We design the neural networks for computing the best rank-one tensor approximation of complex tensors, and prove that the solution of the networks is locally asymptotically stable in the sense of Lyapunov stability theory. This solution also converges to the local optimal solutions of the best complex rank-one tensor approximation. We illustrate our theoretical results via numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, Brauer-type eigenvalue inclusion sets of stochastic tensors and irreducible tensors are given, respectively. Some sufficient conditions of positive (semi-)definiteness of stochastic tensors are obtained. Furthermore, the stability of a high-order nonlinear system is analysed by positive definiteness of tensors. [ABSTRACT FROM AUTHOR]

The total least squares (TLS) represents a popular data fitting approach for solving linear approximation problems A x ≈ b (i.e., with a vector right-hand side) and A X ≈ B (i.e., with a matrix right-hand side) contaminated by errors. This paper introduces a generalization of TLS formulation to problems with structured right-hand sides. First, we focus on the case, where the right-hand side and consequently also the solution are tensors. We show that whereas the basic solvability result can be obtained directly by matricization of both tensors, generalization of the core problem reduction is more complicated. The core reduction allows to reduce mathematically the problem dimensions by removing all redundant and irrelevant data from the system matrix and the right-hand side. We prove that the core problems within the original tensor problem and its matricized counterpart are in general different. Then, we concentrate on problems with even more structured right-hand sides, where the same model A corresponds to a set of various tensor right-hand sides. Finally, relations between the matrix and tensor core problem are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present an equitable partition theorem of tensors, which gives the relations between H -eigenvalues of a tensor and its quotient equitable tensor and extends the equitable partitions of graphs to hypergraphs. Furthermore, with the aid of it, some properties and H -eigenvalues of the generalized power hypergraphs are obtained, which extends some known results, including some results of Yuan, Qi and Shao [20] . [ABSTRACT FROM AUTHOR]

P- and P 0 -matrix classes have wide applications in mathematical analysis, linear and nonlinear complementarity problems, etc., since they contain many important special matrices, such as positive (semi-)definite matrices, M-matrices, diagonally dominant matrices, etc. By modifying the existing definitions of P- and P 0 -tensors that work only for even order tensors, in this paper, we propose a homogeneous formula for the definition of P- and P 0 -tensors. The proposed P- and P 0 -tensor classes coincide the existing ones of even orders and include many important structured tensors of odd orders. We show that many checkable classes of structured tensors, such as the nonsingular M-tensors, the nonsingular H-tensors with positive diagonal entries, the strictly diagonally dominant tensors with positive diagonal entries, are P-tensors under the new definition, regardless of whether the order is even or odd. In the odd order case, our definition of P 0 -tensors, to some extent, can be regarded as an extension of positive semi-definite (PSD) tensors. The theoretical applications of P- and P 0 -tensors under the new definition to tensor complementarity problems and spectral hypergraph theory are also studied. [ABSTRACT FROM AUTHOR]

In this paper, we give upper bounds on Z t -spectral radius of a tensor A ( t = 1 , 2 ) , which extend the upper bounds of Brauer to tensors. Moreover, an upper bound on the Z 1 -spectral radius is proposed via modulus sum of the entries of certain dimension of A , which improves the upper bound given by Li et al. Numerical experiments are given to illustrate the utility of the upper bound. [ABSTRACT FROM AUTHOR]

This paper is devoted to the computation of rank-one decomposition of complex symmetric tensors. Based on the Takagi factorization of complex symmetric matrices, we derive algorithm for computing the partial orthogonal rank-one decomposition of complex symmetric tensors with an order being a power of two, denoted by CSTPOROD . We consider the properties of this decomposition. We design a strategy (tensor embedding) to computing the partial orthogonal rank-one decomposition of complex symmetric tensors, whose order is not the power of two. Similar to the case of complex symmetric tensors, we consider how to compute the partial orthogonal rank-one decomposition of general complex tensors. We illustrate our algorithms via numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, firstly, we introduce the variant tensor splittings, and present some equivalent conditions for a strong M -tensor based on the tensor splitting. Secondly, the existence and uniqueness conditions of the solution for multi-linear systems are given. Thirdly, we propose some tensor splitting algorithms for solving multi-linear systems with coefficient tensor being a strong M -tensor. As an application, a tensor splitting algorithm for solving the multi-linear model of higher order Markov chains is proposed. Numerical examples are given to demonstrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The T-congruence Sylvester equation is the matrix equation A X + X T B = C , where A ∈ R m × n , B ∈ R n × m and C ∈ R m × m are given, and matrix X ∈ R n × m is to be determined. The T-congruence Sylvester equation has recently attracted attention because of a relationship with palindromic eigenvalue problems. For example, necessary and sufficient conditions for the existence and uniqueness of solutions, and numerical solvers have been intensively studied. In this paper, we will show that, under a certain condition and n = m , the T-congruence Sylvester equation can be transformed into the Lyapunov equation. This may lead to further properties and efficient numerical solvers by utilizing the rich literature on the Lyapunov equation. [ABSTRACT FROM AUTHOR]

The tensor rank decomposition problem consists of recovering the parameters of the model from an identifiable low-rank tensor. These parameters are analyzed and interpreted in many applications. As tensors are often perturbed by measurement errors in practice, one must investigate to what extent the unique parameters change in order to preserve the validity of the analysis. The magnitude of this change can be bounded asymptotically by the product of the condition number and the norm of the perturbation to the tensor. This paper introduces a condition number that admits a closed expression as the inverse of a particular singular value of Terracini's matrix, which represents the tangent space to the set of tensors of fixed rank. A practical algorithm for computing this condition number is presented. The latter's elementary properties such as scaling and orthogonal invariance are established. Rank-1 tensors are always well-conditioned. The class of weak 3-orthogonal tensors, which includes orthogonally decomposable tensors, contains both well-conditioned and ill-conditioned problems. Numerical experiments confirm that the condition number yields a good estimate of the magnitude of the change of the parameters when the tensor is perturbed. [ABSTRACT FROM AUTHOR]

This paper presents the neural dynamical network to compute a best rank-one approximation of a real-valued tensor. We implement the neural network model by the ordinary differential equations (ODE), which is a class of continuous-time recurrent neural network. Several new properties of solutions for the neural network are established. We prove that the locally asymptotic stability of solutions for ODE by constructive an appropriate Lyapunov function under mild conditions. Furthermore, we also discuss how to use the proposed neural networks for solving the tensor eigenvalue problem including the tensor H-eigenvalue problem, the tensor Z-eigenvalue problem, and the generalized eigenvalue problem with symmetric-definite tensor pairs. Finally, we generalize the proposed neural networks to the computation of the restricted singular values and the associated restricted singular vectors of real-valued tensors. We illustrate and validate theoretical results via numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is devoted to the extension of the ϵ -pseudo-spectra theory from matrices to tensors. Based on the definition of an eigenpair of real symmetric tensors and results on the ϵ -pseudo-spectrum of square matrices, we first introduce the ϵ -pseudo-spectrum of a complex tensor and investigate its fundamental properties, such as its computational interpretations and the link with the stability of its related homogeneous dynamical system. We then extend the ϵ -pseudo-spectrum to the setting of tensor polynomial eigenvalue problems. We further derive basic structure of the ϵ -pseudo-spectrum for tensor polynomial eigenvalue problems including the symmetry, boundedness and number of connected components under suitable mild assumptions. Finally, we discuss the implication of the ϵ -pseudo-spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial. [ABSTRACT FROM AUTHOR]

Let W be a finite dimensional vector space over a field with characteristic not equal to 2. Denote by V S and V A the subspaces of symmetric and antisymmetric tensors of a subspace V of W ⊗ W , respectively. In this paper we show that if V is generated by tensors with tensor rank 1, V = V S ⊕ V A and W is the smallest vector space such that V ⊂ W ⊗ W then dim ⁡ ( V S ) ≥ max ⁡ { 2 dim ⁡ ( V A ) dim ⁡ ( W ) , dim ⁡ ( W ) 2 } . This result has a straightforward application to the separability problem in Quantum Information Theory: If ρ ∈ M k ⊗ M k ≃ M k 2 is separable then rank ( ( I d + F ) ρ ( I d + F ) ) ≥ max { 2 r rank ( ( I d − F ) ρ ( I d − F ) ) , r 2 } , where M n is the set of complex matrices of order n , F ∈ M k ⊗ M k is the flip operator, I d ∈ M k ⊗ M k is the identity and r is the marginal rank of ρ + F ρ F . We prove the sharpness of this inequality. This inequality is a necessary condition for separability. Moreover, we show that if ρ ∈ M k ⊗ M k is positive under partial transposition (PPT) and rank ( ( I d + F ) ρ ( I d + F ) ) = 1 then ρ is separable. This result follows from Perron–Frobenius theory. We also present a large family of PPT matrices satisfying rank ( I d + F ) ρ ( I d + F ) ≥ r ≥ 2 r − 1 rank ( I d − F ) × ρ ( I d − F ) . There is a possibility that a PPT matrix ρ ∈ M k ⊗ M k satisfying 1 < rank ( I d + F ) ρ ( I d + F ) < 2 r rank ( I d − F ) ρ ( I d − F ) exists. In this case ρ is entangled. This is a gap where we can look for PPT entanglement. [ABSTRACT FROM AUTHOR]

The spectral p -norm of r -matrices generalizes the spectral 2-norm of 2-matrices. In 1911 Schur gave an upper bound on the spectral 2-norm of 2-matrices, which was extended in 1934 by Hardy, Littlewood, and Pólya to r -matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for 2-matrices. The main result of this paper extends the latter result to r -matrices, thereby improving the result of Hardy, Littlewood, and Pólya. The proof is based on combinatorial concepts like r-partite r-matrix and symmetrant of a matrix, which appear to be instrumental in the study of the spectral p -norm in general. Thus, another application shows that the spectral p -norm and the p -spectral radius of a symmetric nonnegative r -matrix are equal whenever p ≥ r . This result contributes to a classical area of analysis, initiated by Mazur and Orlicz back in 1930. Additionally, a number of bounds are given on the p -spectral radius and the spectral p -norm of r -matrices and r -graphs. [ABSTRACT FROM AUTHOR]

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions, namely the higher order singular value decomposition (HOSVD), the tensor train format (TT), and the general hierarchical Tucker decomposition (HT). We provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank. We show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format. These bounds are extended to partial Fourier maps combined with random sign flips of the tensor entries. Finally, we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from Gaussian random measurements, tensor completion (recovery of missing entries), and Fourier measurements for third order tensors. [ABSTRACT FROM AUTHOR]

A Markov chain with memory is no different from the conventional Markov chain on the product state space. Such a Markovianization, however, increases the dimensionality exponentially. Instead, Markov chain with memory can naturally be represented as a tensor, whence the transitions of the state distribution and the memory distribution can be characterized by specially defined tensor products. In this context, the progression of a Markov chain can be interpreted as variants of power-like iterations moving toward the limiting probability distributions. What is not clear is the makeup of the “second dominant eigenvalue” that affects the convergence rate of the iteration, if the method converges at all. Casting the power method as a fixed-point iteration, this paper examines the local behavior of the nonlinear map and identifies the cause of convergence or divergence. As an application, it is found that there exists an open set of irreducible and aperiodic transition probability tensors where the Z -eigenvector type power iteration fails to converge. [ABSTRACT FROM AUTHOR]

Let M n be the algebra of all n × n complex matrices and r ≥ 2 a fixed integer. The aim of this paper is to characterize linear maps ϕ : M m 1 ⋯ m l → M m 1 ⋯ m l such that ϕ ( A 1 ⊗ ⋯ ⊗ A l ) is r -potent whenever A 1 ⊗ ⋯ ⊗ A l is r -potent. [ABSTRACT FROM AUTHOR]

In this paper we characterize additive maps between tensor spaces that send decomposable tensors to decomposable tensors. As an application, we classify all additive maps from tensor products of spaces of rectangular matrices to spaces of rectangular matrices which do not increase the rank of tensor product of rank one matrices. [ABSTRACT FROM AUTHOR]

We extend to hypermatrices definitions and theorem from matrix theory. Our main result is an elementary derivation of the spectral decomposition of hypermatrices generated by arbitrary combinations of Kronecker products and direct sums of cubic side length 2 hypermatrices. The method is based on a generalization of Parseval's identity. We use this general formulation of Parseval's identity to introduce hypermatrix Fourier transforms and discrete Fourier hypermatrices. We extend to hypermatrices a variant of the Gram–Schmidt orthogonalization process as well as Sylvester's classical Hadamard matrix construction. We conclude the paper with illustrations of spectral decompositions of adjacency hypermatrices of finite groups and a short proof of the hypermatrix formulation of the Rayleigh quotient inequality. [ABSTRACT FROM AUTHOR]

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended to hypermatrices. To this end, the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric r -matrix A has a symmetric spectrum if and only if r is even and A is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs with symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu. Separately, similar results are obtained for the H -spectrum of hypermatrices. [ABSTRACT FROM AUTHOR]

This paper proposes complex-valued neural network for computing the Takagi vectors corresponding to the largest Takagi value of complex symmetric matrices. We establish some properties of the complex-valued neural network. Based on the Takagi factorization of complex symmetric matrices, we establish an explicit representation for the solution of the neural network and analyze its convergence property. Under certain conditions, we design a strategy to computing the Takagi factorization of a complex symmetric matrix by the proposed neural network. As an application, we consider the left and right singular vectors associated with the largest singular value for complex Toeplitz matrices. We illustrate our theory via numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we give two Brauer-type eigenvalue inclusion sets and some bounds on the spectral radius for tensors. As applications, some bounds on the spectral radius of uniform hypergraphs are presented. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to generalize matrix numerical ranges to the tensor case based on tensor norms. We show that the basic properties of matrix numerical ranges such as compactness and convexity are valid for tensor numerical ranges. We make use of convexity property to propose an algorithm for approximating tensor numerical ranges in which tensor eigenvalues are contained. Also we consider tensor numerical ranges based on inner products, however, they may not be convex in general. [ABSTRACT FROM AUTHOR]

Identifying the positive definiteness of an even-order homogeneous multivariate form is an important task due to its wide applications in such as medical imaging and the stability analysis of nonlinear autonomous systems via Lyapunov’s direct method in automatic control and multivariate network realizability analysis. In this paper, based on the equivalence of the positive definiteness of the form to that of the underlying tensor, and the links between the positive definiteness of a tensor with strong H -tensor, we propose an H -tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. The validity of the iterative scheme is guaranteed theoretically and the given numerical experiments show the efficiency of the scheme. [ABSTRACT FROM AUTHOR]

For a tensor A = ( a i 1 ⋯ i m ) ∈ C n × ⋯ × n , the associated digraph of A has the vertex set V ( A ) = { 1 , … , n } , and the arc set of Γ A is E ( A ) = { ( i , j ) | a i i 2 ⋯ i m ≠ 0 , j ∈ { i 2 , … , i m } ≠ { i , … , i } } . In this paper, we give some characterizations of M -tensors and H -tensors by using the associated digraph of a tensor. [ABSTRACT FROM AUTHOR]

In this paper, we consider the left (right) inverse of a tensor. We characterize the existence of any order k left (right) inverse of a tensor, and show the expression of left (right) inverse of a tensor. We also present a result for the similarity of tensors. [ABSTRACT FROM AUTHOR]

In this paper we study the tensor product for MV-algebras, the algebraic structures of Łukasiewicz ∞-valued logic. Our main results are: the proof that the tensor product is preserved by the categorical equivalence between the MV-algebras and abelian lattice-order groups with strong unit and the proof of the scalar extension property for semisimple MV-algebras. We explore consequences of these results for various classes of MV-algebras and lattice-ordered groups enriched with a product operation. [ABSTRACT FROM AUTHOR]

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors { x ( i ) } with x ( i ) ∈ R n 1 × ⋯ × n d in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram–Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α i in an arbitrary tensor v = ∑ i α i x ( i ) . The orthonormal Q factor in the QR decomposition X ≡ [ x ( 1 ) , ⋯ , x ( p ) ] = Q R cannot be computed but expressed as X R − 1 when required. The resulting algorithm has an O ( p 2 d n ) computational complexity, with n = max ⁡ n i . Some illustrative examples in the numerical solution of tensor linear equations are presented. [ABSTRACT FROM AUTHOR]

In this paper, we carefully examine the structure of the gradient of an operator norm on a finite-dimensional matrix space. In particular, we derive concise and useful representations for an operator norm and its subgradient, which refine existing results in this area of study. We further use the derived representations to formulate and prove a criterion of an operator norm, the first of its kind, to the best of our knowledge. It essentially states that a matrix norm is an operator norm if and only if the set of its gradients is the set of the outer products of vectors from each pair of the Cartesian product of two vector sets . We also provide several handy tests, based on this criterion, which in certain cases help to determine whether a matrix norm is an operator norm or not. In addition, we generalize our theoretical developments to higher dimensions, i.e. for injective norms on tensor spaces. [ABSTRACT FROM AUTHOR]

Research in tensor representation and analysis has been rising in popularity in direct response to a) the increased ability of data collection systems to store huge volumes of multidimensional data and b) the recognition of potential modeling accuracy that can be provided by leaving the data and/or the operator in its natural, multidimensional form. In recent work [1] , the authors introduced the notion of the t-product, a generalization of matrix multiplication for tensors of order three, which can be extended to multiply tensors of arbitrary order [2] . The multiplication is based on a convolution-like operation, which can be implemented efficiently using the Fast Fourier Transform (FFT). The corresponding linear algebraic framework from the original work was further developed in [3] , and it allows one to elegantly generalize all classical algorithms from numerical linear algebra. In this paper, we extend this development so that tensor–tensor products can be defined in a so-called transform domain for any invertible linear transform. In order to properly motivate this transform-based approach, we begin by defining a new tensor–tensor product alternative to the t-product. We then show that it can be implemented efficiently using DCTs, and that subsequent definitions and factorizations can be formulated by appealing to the transform domain. Using this new product as our guide, we then generalize the transform-based approach to any invertible linear transform. We introduce the algebraic structures induced by each new multiplication in the family, which is that of C ⁎ -algebras and modules. Finally, in the spirit of [4] , we give a matrix–algebra based interpretation of the new family of tensor–tensor products, and from an applied perspective, we briefly discuss how to choose a transform. We demonstrate the convenience of our new framework within the context of an image deblurring problem and we show the potential for using one of these new tensor–tensor products and resulting tensor-SVD for hyperspectral image compression. [ABSTRACT FROM AUTHOR]

Non-Orthogonal Joint Diagonalization (NOJD) of a given real symmetric matrix set A = { A j } j = 0 p is to find a nonsingular matrix W such that W ⊤ A j W for j = 0 , 1 , … , p are all as diagonal as possible. If the columns of the solution W are all required to be unit length, we call such NOJD problem as the Normalized NOJD (NNOJD) problem. In this paper, we discuss the perturbation theory for NNOJD as an optimization problem. Based on the perturbation analysis of general constrained optimization problem given in [16] , we obtain an upper bound for the distance between an approximated solution of the perturbed optimal problem and the set of exact joint diagonalizers. As corollaries, a perturbation bound and an error bound are also given. Numerical examples validate the bounds. [ABSTRACT FROM AUTHOR]

In this paper, we define the sign pattern tenors, minimum (maximum) rank of sign pattern tenors, term rank of tensors and sign nonsingular tensors. The necessity and sufficiency for the minimum rank of sign pattern tenors to be 1 is given. We show that the maximum rank of a sign pattern tensor is not less than the term rank and the minimum rank of the sign pattern of a sign nonsingular tensor is not less than its dimension. We get some characterizations of tensors having sign left or sign right inverses. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Z -eigenpair of a tensor, in particular, an irreducible nonnegative tensor. We present some bounds for the eigenvector and Z -spectral radius. The proposed bounds improve some existing ones. An example with practical applications is given to show the proposed bound. [ABSTRACT FROM AUTHOR]

In this paper, a new product of tensors is given, which is a generalization of the product of tensors introduced by Gelfand, Kapranov, Zelevinsky. This product satisfies the associative law. Using the associative law, it is concluded that the product of tensors given by Bu and others and the general product of tensors introduced by Shao can be represented by this new tensors product. As an application of this new tensors product and the associative law, the hyperdeterminant of the product of a kind of tensors is studied, i.e., some formulas for the hyperdeterminant of boundary formats product are shown. [ABSTRACT FROM AUTHOR]

In this paper, we give a necessary and sufficient condition for an even order three dimensional strongly symmetric circulant tensor to be positive semi-definite. We show that this condition can be a sufficient condition for such a tensor to be sum-of-squares in some cases. There are no PNS strongly symmetric circulant tensors to be found in numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, we show that the primitive degree set of nonnegative primitive tensors with order m ( ≥ 3 ) and dimension n is { 1 , 2 , … , ( n − 1 ) 2 + 1 } , which implies that the results of the case m ≥ 3 (the case of tensors) is totally different from the case m = 2 (the case of matrices), and we also propose some open problems for further research. [ABSTRACT FROM AUTHOR]

Primitivity is an important concept in the spectral theory of nonnegative matrices and tensors. It is well-known that an irreducible matrix is primitive if and only if the greatest common divisor of all the cycles in the associated directed graph is equal to 1. The main aim of this paper is to establish a similar result, i.e., we show that a nonnegative tensor is primitive if and only if the greatest common divisor of all the cycles in the associated directed hypergraph is equal to 1. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic not 2 and M n the algebra of all n × n matrices over F . The aim of this paper is to characterize linear maps ϕ : M m 1 ⋯ m l → M m 1 ⋯ m l such that ϕ ( A 1 ⊗ ⋯ ⊗ A l ) is an idempotent whenever A 1 ⊗ ⋯ ⊗ A l is an idempotent. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we provide some interested operator inequalities related with non-negative linear maps by means of concavity and convexity structure, and also establish some new attractive inequalities for the Khatri–Rao products of two or more positive definite matrices. These results lead to inequalities for Hadamard product and Ando’s and α-power geometric means, as a special case. [Copyright &y& Elsevier]

Abstract: Many important spectral properties of nonnegative matrices have recently been successfully extended to higher order nonnegative tensors; for example, see (Chang et al., 2008, 2011; Friedland et al., in press; Lim, 2005; Liu et al., 2010; Ng et al., 2010; Qi et al., 2007; Yang and Yang, 2010) [2,3,9,17,23,24,27,28]. However, most of these results focus on the H-eigenvalues introduced by Qi (2005, 2007) [25,26]. The key results of this paper reveal some similarities as well as some crucial differences between Z-eigenvalues and H-eigenvalues of a nonnegative tensor. In particular, neither the positive Z-eigenvalue nor the associated positive Z-eigenvector of an irreducible nonnegative tensor has to be unique in general as demonstrated by Example 2.7. Furthermore, the Collatz type min–max characterizations of the largest positive Z-eigenvalue of an irreducible nonnegative tensor is only partially true in general as seen in Theorem 4.7 and Example 4.8. [Copyright &y& Elsevier]