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We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well-known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic $n\times n$ matrices. We establish that in case $0, Comment: Version 2: some proofs are more detailed; to appear in International Journal of Algebra and Computation

Let V be an n-dimensional algebraic representation over an algebraically closed field K of a group G. For m > 0, we study the invariant rings K[Vm]G for the diagonal action of G on Vm. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in K[Vm]G by the process of polarization and restitution from K[Vn]G. In particular, this means that if K[Vn]G is generated in degree ≤ d, then so is K[Vm]G no matter how large m is. There are several explicit counterexamples to Weyl’s theorem in positive characteristic. However, when G is a (connected) reductive affine group scheme over ℤ and V*is a good G-module, we show that Weyl's theorem holds in sufficiently large characteristic. As a special case, we consider the ring of invariants R(n, m) for the left-right action of SLn × SLn on m-tuples of n × n matrices. In this case, we show that the invariants of degree ≤ n6 suffice to generate R(n, m) if the characteristic is larger than 2n6 + n2.

Betti curves of symmetric matrices were introduced in [3] as a new class of matrix invariants that depend only on the relative ordering of matrix entries. These invariants are computed using persistent homology, and can be used to detect underlying structure in biological data that may otherwise be obscured by monotone nonlinearities. Here we prove three theorems that characterize the Betti curves of rank 1 symmetric matrices. We then illustrate how these Betti curve signatures arise in natural data obtained from calcium imaging of neural activity in zebrafish.

International audience; We uncover new behaviors of the transmission of information by three quantum channels in superposition of causal orders subject to some level of noise. We find that the transmission can exhibit three different behaviors as the level of noise is varied. This multifold behavior can be explained by the different equivalence classes of quantum switch matrices related to specific combinations of causal orders. We classify these matrices using their characteristic polynomials and matrix invariants, and we calculate analytical expressions for the Holevo information in three representative cases. Our results are a step forward to understand and harness quantum control of causal orders with different levels of noise. We also study the Holevo information as function of a continuous order parameter and analyse transitions at integer values.

A decomposition of the magnetotelluric tensor is described in terms of quantities which are invariant to the rotation of observing axes, and which also are distinct measures of the 1D, 2D or 3D characteristics of the tensor and so may be useful in dimensionality analysis. When the in-phase and quadrature parts of the tensor are analysed separately there are two invariants which gauge 1D structure, two invariants which gauge 2D structure, and three invariants which gauge 3D structure. A matrix method similar to singular value decomposition is used to determine many of the invariants, and their display is then possible on Mohr diagrams. A particular set of invariants proposed some seventeen years ago is revised to yield an improved set. Several possibilities for the seventh invariant are canvassed, and illustrated by examples from field data. Low values of Δβ, the invariant now preferred for ‘the 7th’, may indicate a particular simplification of otherwise complicated three-dimensional structure.

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index [math] in terms of the maximal degree in a minimal homogenous generating system of the ring of simultaneous conjugation invariants of tuples of [math] -by- [math] matrices. This is deduced from a result of Zubkov. As a consequence, a recent degree bound due to Derksen and Makam for the generators of the ring of matrix invariants yields an upper bound for the nilpotency index of a finitely generated nil algebra that is polynomial in the number of generators and the nil index. Furthermore, a characteristic free treatment is given to Kuzmin’s lower bound for the nilpotency index.

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.

It is well known that the ring of polynomial invariants of a reductive group is finitely generated. However, it is difficult to give strong upper bounds on the degrees of the generators, especially over fields of positive characteristic. In this paper, we make use of the theory of good filtrations along with recent results on the null cone to provide polynomial bounds for matrix semi-invariants in arbitrary characteristic, and consequently for matrix invariants. Our results generalize to invariants and semi-invariants of quivers.

The objective (Lagrangian and Eulerian) strain tensors, their rates, and conjugate stress tensors used in continuum mechanics equations are considered. These tensors are represented in the eigenprojections of the right and left stretch tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The novelty of the research lies in the simultaneous derivation of expressions for Lagrangian versions of tensors and their Eulerian counterparts with decomposition of the obtained expressions into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . We consider the Lagrangian and Eulerian Doyle–Ericksen families of strain tensors, which, in turn, are subfamilies of the well-known Hill family of strain tensors. Basis-free expressions are obtained for the material rates of Lagrangian tensors from the previously introduced family of generalized strain tensors, whose subfamilies are the Lagrangian and Eulerian Hill families of strain tensors, and similar expressions are obtained for the Green–Naghdi rates of Eulerian strain tensors, which are Eulerian counterparts of the material rates of Lagrangian strain tensors. Basis-free expressions for stress tensors are derived from the classical definitions of the conjugacy of stress and strain tensors. Expressions are found for the Lagrangian and Eulerian symmetric Atluri stress tensors that do not have conjugate strain tensors from the Hill family, and the obtained expressions are decomposed into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The ranges of allowed ratios of different principal stretches at which the Lagrangian and Eulerian Hencky and Atluri stress tensors approximate the rotated/standard Kirchhoff stress tensors are determined. It is shown that the range of allowed ratios for the Hencky stress tensors is wider than that for the Atluri stress tensors.

Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares (SOS) algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors (EEN-tensors), which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a nonnegative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.

In this paper, we extend some well-known results on the relationships of the eigenvalues and eigenvectors from matrices to the tensors case, including nonsingular tensors and their left (right) inverses, similar tensors, tensor products and , where B is a matrix.

An S-type singular value inclusion set for rectangular tensors is given. Based on the set, new upper and lower bounds for the largest singular value of nonnegative rectangular tensors are obtained and proved to be sharper than some existing results. Numerical examples are given to verify the theoretical results.

This paper is devoted to the computation of US-eigenpairs of complex symmetric tensors and U-eigenpairs of complex tensors. Based on the Takagi factorization of complex symmetric matrices, we derive an iterative algorithm for computing US-eigenpairs of complex symmetric tensors, denoted as QRCST Algorithm. We also observe that multiple US-eigenpairs can be found from a local permutation heuristic, which is effectively a tensor similarity transformation, resulting in the permuted version of QRCST. We then generalize our techniques to general complex tensors. Finally, we derive a higher order power type method for computing a US- or a U-eigenpair, similar to the higher-order power method for computing Z-eigenpairs of real symmetric tensors or a best rank-one approximation of real tensors. We illustrate our algorithms via numerical examples.

In this paper, we mainly focus on new inclusion sets for eigenvalues of a tensor. First, we propose new inclusion sets for eigenvalues of a tensor, which are sharper than some existing inclusion sets, and obtain the law of distribution of the number of eigenvalues for a tensor. Second, two new classes of tensors are introduced. Third, some bounds on the spectral radii for nonnegative tensors are given. Fourth, some checkable sufficient conditions for the positive definiteness (positive semidefiniteness) of some classes of even-order real symmetric tensors are obtained.

A new Z-eigenvalue localization set for tensors is given and proved to be tighter than those in the work of Wang et al. (Discrete Contin. Dyn. Syst., Ser. B 22(1):187-198, 2017). Based on this set, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

A new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensor are obtained. It is proved that the new lower and upper bounds improve the corresponding bounds provided by He and Huang (J. Inequal. Appl., 2014, 2014, 114) and Zhao and Sang (J. Inequal. Appl., 2016, 2016, 268). Finally, two numerical examples are given to verify the theoretical results.

The algebra of invariants of d-tuples of skew-symmetric matrices under the action of the orthogonal group O(n) by the simultaneous conjugation is considered over an infinite field of characteristic...

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (M) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (M) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.

In this paper, we consider the eigenvalue problem for a tensor of arbitrary even rank. In this connection, we state definitions and theorems related to the tensors of moduli ℂ2p(Ω) and ℝ2p(Ω), where p is an arbitrary natural number and Ω is a domain of the n-dimensional Riemannian space ℝn. We introduce the notions of minor tensors and extended minor tensors of rank (2ps) and order s, the corresponding notions of cofactor tensors and extended cofactor tensors of rank (2ps) and order (N−s), and also the cofactor tensors and extended cofactor tensors of rank 2p(N−s) and order s for rank-(2p) tensor. We present formulas for calculation of these tensors through their components and prove the Laplace theorem on the expansion of the determinant of a rank-(2p) tensor by using the minor and cofactor tensors. We also obtain formulas for the classical invariants of a rank-(2p) tensor through minor and cofactor tensors and through first invariants of degrees of a rank-(2p) tensor and the inverse formulas. A complete orthonormal system of eigentensors for a rank-(2p) tensor is constructed. Canonical representations for the specific strain energy and determining relations are obtained. A classification of anisotropic linear micropolar media with a symmetry center is proposed. Eigenvalues and eigentensors for tensors of elastic moduli for micropolar isotropic and orthotropic materials are calculated.

In this paper, we give a note on the eigenvalue localization sets for tensors. We show that these sets are tighter than those provided by Li et al. (2014) [1].

This paper contains structural and logical scheme of the research; theoretical information about the set of azimuthally invariant Mueller-matrix elements and their combinations; The work is aimed at the development of a set of techniques that form a new method of azimuthally invariant differential polarimetry of partially-depolarizing optically anisotropic biological layers. This method is based on the determination and diagnostic use of a set of physical relationships between the distributions of azimuthally invariant polarization parameters characterizing the optical anisotropy of partially depolarizing layers of biological tissues, and the distributions of the parameters of linear and circular birefringence of such objects.

We give a further study on B-tensors and introduce doubly Btensors that contain B-tensors. We show that they have similar properties, including their decompositions and strong relationship with strictly (doubly) diagonally dominated tensors. As an application, the properties of B-tensors are used to localize real eigenvalues of some tensors, which would be very useful in verifying the positive semi-definiteness of a tensor.

Transition probability tensors of order 3 in dimension 3 and of order 4 in dimension 2 are studied. In both cases, we prove that an irreducible symmetric transition probability tensor has a unique positive -eigenvector.

We first introduced a linear stationary equation with a quadratic operator in ∂ x and ∂ y , then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

International audience; The orthogonalized harmonic decomposition of symmetric fourth-order tensors (i.e. having major and minor indicial symmetries, such as elasticity tensors) is completed by a representation of harmonic fourth-order tensors HH by means of two second-order harmonic (symmetric deviatoric) tensors only. A similar decomposition is obtained for non-symmetric tensors (i.e. having minor indicial symmetry only, such as photo-elasticity tensors or elasto-plasticity tangent operators) introducing a fourth-order major antisymmetric traceless tensor ZZ. The tensor ZZ is represented by means of one harmonic second-order tensor and one antisymmetric second-order tensor only. Representations of totally symmetric (rari-constant), symmetric and major antisymmetric fourth-order tensors are simple particular cases of the proposed general representation. Closed-form expressions for tensor decomposition are given in the monoclinic case. Practical applications to elasticity and photo-elasticity monoclinic tensors are finally presented.; La décomposition harmonique orthogonalisée des tenseurs symétriques d'ordre quatre (ayant les symétries majeures et mineures, tels que le tenseur d'élasticité) est complétée par une représentation des tenseurs harmoniques d'ordre quatre HH à l'aide de deux tenseurs harmoniques (symétriques déviatoriques) d'ordre deux. Une décomposition similaire est obtenue pour les tenseurs non symétriques (ayant uniquement la symétrie mineure, tels que ceux rencontrés en photo-élasticité et en élasto-plasticité), introduisant un tenseur antisymétrique majeur à traces nulles ZZ. Le tenseur ZZ est représenté par deux tenseurs d'ordre deux, le premier harmonique et le second antisymétrique. Les représentations des tenseurs d'ordre quatre complètement symétriques (rari-constants), symétriques et antisymétriques majeurs sont des cas particuliers simples de la représentation proposée. Les expressions analytiques de la décomposition correspondante dans le cas monoclinique sont obtenues et appliquées à l'élasticité et à la photo-élasticité monocliniques.

To investigate complex physical phenomena, bi-dimensional models are often an interesting option. It allows spatial couplings to be produced while keeping them as simple as possible. For linear physical laws, constitutive equations involve the use of tensor spaces. As a consequence the different types of anisotropy that can be described are encoded in tensor spaces involved in the model. In the present paper, we solve the general problem of computing symmetry classes of constitutive tensors in [Formula: see text] using mathematical tools coming from representation theory. The power of this method is illustrated through the tensor spaces of Mindlin strain-gradient elasticity.

Several new criteria for identifying -tensors are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Advantages of the obtained results are illustrated by numerical examples.

The admitted region of five joint irreducible invariants of the strain and rotation rate tensors calculated as the traces of the products of these tensors is determined.

We consider tensors of format over the finite field . We use computer algebra to classify these tensors by their tensor rank, thus determining the maximum tensor rank to be 9. As a corollary, we provide a new upper bound that the maximum rank of an order-n tensor of format , for , over is at most . We also determine that there are 261 canonical forms of the rank 9 (maximum rank) tensors under the action of , the semi-direct product of (a direct product of) general linear groups with the symmetric group on five elements.

A closed form expression is presented for the irreducible Cartesian tensor of highest weight, for arbitrary order. Two proofs are offered, one employing bookkeeping of indices and, after establishing the connection with the so-called natural tensors and their projection operators, the other one employing purely coordinate-free tensor manipulations. Some theorems and formulas in the published literature are generalized from SO(3) to SO( n ), for dimensions n ≥ 3 .

Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly evenbipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Ztensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii.

New iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.

The completely positive (CP) tensor verification and decomposition are essential in tensor analysis and computation due to the wide applications in statistics, computer vision, exploratory multiway data analysis, blind source separation, and polynomial optimization. However, it is generally NP-hard as we know from its matrix case. To facilitate the CP tensor verification and decomposition, more properties for the CP tensor are further studied, and a great variety of its easily checkable subclasses such as the positive Cauchy tensors, the symmetric Pascal tensors, the Lehmer tensors, the power mean tensors, and all of their nonnegative fractional Hadamard powers and Hadamard products are exploited in this paper. Particularly, a so-called CP-Vandermonde decomposition for positive Cauchy--Hankel tensors is established and a numerical algorithm is proposed to obtain such a special type of CP decomposition. The doubly nonnegative (DNN) matrix is generalized to higher-order tensors as well. Based on the DNN ten...

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie’s finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allows to compute differential invariants from joint invariants.

In this work we derive important properties regarding matrix invariants which occur in the theory of differential equations with reflection., Preprint

We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give efficient algorithms to decide if the orbit closures of two points intersect. We also improve the known bounds for the degree of separating invariants in these cases., Better bounds for separating invariants and improved exposition in some sections

In this chapter we show that when E\(_{\textit{n}}\) is a Euclidean vector space, there is an isomorphism among the tensor spaces T\(_{\textit{s}}^{\textit{r}}\)(E\(_{\textit{n}}\)) for which r+s has a given value. In other words, we show the existence of an isomorphism between \(\textit{E}_{\textit{n}}\) and \(\textit{E}_{\textit{n}}^{{*}}\), of isomorphisms between T\(_{0}^{2}\), T\(_{1}^{1}\), and T\(_{2}^{0}\), and so on. These isomorphisms lead to the definitions of Euclidean vectors, tensors, and to the covariant, contravariant, and mixed components of a tensor.

Various tensor formats are used for the data-sparse representation of large-scale tensors. Here we investigate how symmetric or antisymmetric tensors can be represented. We mainly investigate the hierarchical format, but also the use of the canonical format is mentioned.

A generalization of the complex moments from 2D to 3D is described. Group representation theory is used to construct 3D rotation invariants from them. The algorithm for automatic generating of the invariants of higher orders is proposed. An algorithm for automatic generation of higher order invariants is proposed. The linearly dependent invariants are eliminated. The invariants are experimentally tested on 3D graphical models and also on real volumetric data.

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.

In this paper, we define the Moore–Penrose inverse of tensors with the Einstein product, and the explicit formulas of the Moore–Penrose inverse of some block tensors are obtained. The general solutions of some multilinear systems are given and we also give the minimum-norm least-square solution of some multilinear systems using the Moore–Penrose inverse of tensors.

A new tensor eigenvalue inclusion set is given, and proved to be tighter than those in L.Q. Qi (2005) [18] and C.Q. Li, Y.T. Li, X. Kong (2014) [12] . In addition, we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by Qi (2005) for weakly irreducible tensors. As applications, we give new bounds for the spectral radius of nonnegative tensors, some sufficient conditions for a tensor to be a strong M -tensor and some inequalities to identify the positive definiteness for an even-order real supersymmetric tensor.

By using digraphs of tensors, we give Brualdi-type eigenvalue inclusion sets of tensors. We also give some applications of our result to nonsingularity and positive definiteness of tensors.

In this paper, we study an eigenvalue problem for even order tensors. Using the matrix unfolding of even order tensors, we can establish the relationship between a tensor eigenvalue problem and a multilevel matrix eigenvalue problem. By considering a higher order singular value decomposition of a tensor, we show that higher order singular values are the square root of the eigenvalues of the product of the tensor and its conjugate transpose. This result is similar to that in matrix case. Also we study an eigenvalue problem for Toeplitz/circulant tensors, and give the lower and upper bounds of eigenvalues of Toeplitz tensors. An application in image restoration is also discussed.

We derive simple integral solutions for the Green’s tensors and Eshelby tensor for a generally anisotropic and non-centrosymmetric micropolar material. The material properties of a micropolar medium are characterized by three fourth-order tensors, C , B and D , in which C relates the stress to the strain, D connects the couple stress to the curvature tensor, and B is the coupling tensor, linking the stress to the curvature tensor or the couple stress to the strain tensor. Here we find that when C and B tensors possess minor symmetry conditions, the Green’s tensors for a general anisotropic micropolar medium can be derived in simple integral expressions. We note however that in our formulation there is no any restriction on the D tensor. Specifically, we will show that, under the conditions on the C and B tensors, the Green’s tensors can be exactly expressed as a simple line integral over a unit circle, and that the Eshelby’s tensors for a general ellipsoidal inclusion can be derived as surface integrals over a unit sphere, entirely analogous to those of the classical elasticity. The exact integral form of the tensors can be implemented with numerical integration procedures. We will demonstrate that our numerical solutions are in good accuracy compared with the existing solutions for simple situations. In the literature analytic Green’s tensors are existed only for isotropic and non-centrosymmetric micropolar medium. The solutions derived here can serve as benchmark solutions for certain classes of anisotropic micropolar medium, and also can be used as an approximate solution for a micropolar medium with general material properties without any constraint conditions.

The main purpose of this paper was to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as symmetric positive-definite tensors and copositive tensors, we derive the existence theorems on a solution of these kinds of nonlinear complementarity problems. We prove that a unique solution of the NCP exists under the condition of diagonalizable tensors.

A new eigenvalue localization set for tensors is given, and proved to be tighter than those in [Qi L. Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 2005;40:1302–1324] and [Li CQ, Li YT, Kong X. New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 2014;21:39–50]. Based on this set, we give two checkable sufficient conditions of the positive definiteness of tensors, and two checkable sufficient conditions of the positive semi-definiteness of tensors.

This article considers the problem of generating polynomial invariants for iterative loops with loop initialization statements and nonsingular linear operators in loop bodies. The set of such invariants forms an ideal in the ring of polynomials in the loop variables. Two algorithms are presented one of which calculates basic invariants for a linear operator in the form of a Jordan cell and the other calculates basic invariants for a diagonalizable linear operator with an irreducible minimal characteristic polynomial. The following theorem on the structure of the basis of the ideal of invariants for such an operator is proved: this basis consists of basic invariants of Jordan cells and basic invariants of the diagonalizable part of the linear operator being considered.