An eigenvalue problem for even order tensors with its applications

Abstract

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.

Multilinear Time Invariant System Theory

“…Brazell et al [7] investigated properties for even-order non-paired tensors A ∈ R J1×···×JN ×I1×···×IN under the Einstein product through construction of an isomorphism to GL(R) (general linear group). The existence of the isomorphism enables one to generalize several matrix concepts, such as orthogonality, invertibility and eigenvalue decomposition to the tensor case [7,14,15,16,17]. We establish an analogous isomorphism for even-order paired tensors by a permutation of indices.…”

E-eigenvalue inclusion theorems for tensors

“…Due to various new and important applications of E-eigenvalue problem in numerical multilinear algebra [21], image processing [22], higher order Markov chains [23], spectral hypergraph theory, the study of quantum entanglement, and so on, some properties of E-eigenvalues have been studied systematically; see [8] for details. However, characterizations of inclusion set for E-eigenvalue are still underdeveloped.…”

Numerical subspace algorithms for solving the tensor equations involving Einstein product

“…Different from the tensor equations (1)–(3), another type of tensor equations is based on the Einstein product. The tensor equation via the Einstein product comes from engineering, the isotropic and anisotropic elastic models, 14 the brain MRI images restoration, 15,16 the biomedical science, 17 the finite element, 18 the finite difference or the spectral method 9,12 and plays very vital role in the discretization of some linear partial differential equations in high dimension. Such as, in physics, for noncentrosymmetric materials, the linear piezoelectric equation is expressed as 19 where is a piezoelectric tensor, is a stress tensor, and is the electric change density displacement.…”

“…The three‐dimensional model of image formation can be expressed as the convolution of an object and a point spread function associated with the noise 20 where G ( x , y , z ) is the image, F ( x , y , z ) is the object, N ( x , y , z ) is the noise, and ∘ denotes the three‐dimensional convolution operator. Using the tensor representation, we can rewritten (5) as 15 where , , and are third‐order tensor representations for the image, the object, and noise functions, respectively, and is the sixth‐order Toeplitz tensor (convolution operator) obtained from a third‐order tensor , that is, …”