On the Representation of Symmetric Isotropic 4-Tensors (original) (raw)
International Journal of Pure and Apllied Mathematics, 2014
In a 4-dimensional Euclidean space, representation theorems have been recently obtained for isotropic functions depending on an arbitrary number of scalars, skew-symmetric second order tensors and symmetric second order tensors; the cases has been treated where at least one of these last ones has an eigenvalue with multiplicity 1 or two distinct eigenvalues with multiplicity 2. The case with at least a non null vector, among the independent variables, was already treated in literature. There remain the case where every symmetric tensor has an eigenvalue with multiplicity 4; but, in this case, it plays a role only through its trace. Consequently, it remains the case where the independent variables, besides scalars, are skew-symmetric tensors. This case is treated in the present paper. As in the other cases, the result is a finite set of scalar valued isotropic functions such that every other scalar function of the same variables can be expressed as a function of the elements of this set. Similarly, a set of tensor valued isotropic functions is found such that every other tensor valued function of the same variables can be expressed as a linear combination, trough scalar coefficients, of the elements of this set. This result is achieved both for symmetric functions , and for skew-symmetric functions.
On the representation theorem for linear, isotropic tensor functions
Journal of Elasticity, 1995
The well-know representation theorem for the elasticity tensor C of an isotropic body shows that C[E] = 2#E + ~ tr(E)I (1) for all symmetric tensors E, where tr(E) denotes the trace of E and I is the identity tensor. This theorem is actually a special case of a classical result (cf. e.g. [Je 31, Chapter 7]) on linear, tensor-valued mappings that are isotropic, i.e. C[QHQ T] = QC[HjQ T for all tensors H in the domain of C and all orthogonal tensors Q, where Q~ denotes the transpose of Q.
Intrinsic characterization of space-time symmetric tensors
Journal of Mathematical Physics, 1992
This paper essentially deals with the classification of a symmetric tensor on a fourdimensional Lorentzian space. A method is given to find the algebraic type of such a tensor. A system of concomitants of the tensor is constructed, which allows one to know the causal character of the eigenspace corresponding to a given eigenvalue, and to obtain covariantly their eigenvectors. Some algebraic as well as differential applications are considered.
On the algebraic structure of second order symmetric tensors in 5-dimensional space-times
General Relativity and Gravitation, 1996
A new approach to the algebraic classification of second order symmetric tensors in 5-dimensional space-times is presented. The possible Segre types for a symmetric two-tensor are found. A set of canonical forms for each Segre type is obtained. A theorem which collects together some basic results on the algebraic structure of the Ricci tensor in 5-dimensional space-times is also stated.
Segre types of symmetric two-tensors inn-dimensional spacetimes
General Relativity and Gravitation, 1995
Three propositions about Jordan matrices are proved and applied to algebraically classify the Ricci tensor in n-dimensional Kaluza-Klein-type spacetimes. We show that the possible Segre types are [1, 1 . . . 1], [21 . . . 1], [31 . . . 1], [zz1 . . . 1] and degeneracies thereof. A set of canonical forms for the Segre types is obtained in terms of semi-null bases of vectors.
Symmetric Tensors and Symmetric Tensor Rank
SIAM Journal on Matrix Analysis and Applications, 2008
A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1.
On second-order, divergence-free tensors
Journal of Mathematical Physics, 2014
This paper deals with the problem of describing the vector spaces of divergence-free, natural tensors on a pseudo-Riemannian manifold that are second-order; i.e., that are defined using only second derivatives of the metric. The main result establish isomorphisms between these spaces and certain spaces of tensors (at a point) that are invariant under the action of an orthogonal group. This result is valid for tensors with an arbitrary number of indices and symmetries among them and, in certain cases, it allows to explicitly compute basis, using the theory of invariants of the orthogonal group. In the particular case of tensors with two indices, we prove the Lovelock tensors are a basis for the vector space of second-order tensors that are divergence-free, thus refining the original Lovelock's statement.
A new class of non-identifiable skew-symmetric tensors
Annali di Matematica Pura ed Applicata (1923 -)
We prove that the generic element of the fifth secant variety σ 5 (Gr(P 2 , P 9)) ⊂ P(3 C 10) of the Grassmannian of planes of P 9 has exactly two decompositions as a sum of five projective classes of decomposable skew-symmetric tensors. We show that this, together with Gr(P 3 , P 8), is the only non-identifiable case among the non-defective secant varieties σ s (Gr(P k , P n)) for any n < 14. In the same range for n, we classify all the weakly defective and all tangentially weakly defective secant varieties of any Grassmannians. We also show that the dual variety (σ 3 (Gr(P 2 , P 7))) ∨ of the variety of 3-secant planes of the Grassmannian of P 2 ⊂ P 7 is σ 2 (Gr(P 2 , P 7)) the variety of bi-secant lines of the same Grassmannian. The proof of this last fact has a very interesting physical interpretation in terms of measurement of the entanglement of a system of 3 identical fermions, the state of each of them belonging to a 8-th dimensional "Hilbert" space.
Tensor Decompositions and Their Properties
Mathematics, 2023
In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ = 1, 2, 3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.