Physics

Tensor contraction

Tensor (intrinsic definition)

Filtered algebra

Riemann curvature tensor

Symmetric tensor

Pure mathematics

Mathematics

Mathematical physics

Multilinear algebra

Multilinear map

Tensor product of Hilbert spaces

Algebra over a field

Tensor density

Mathematical analysis

Division algebra

Einstein tensor

Einstein

Eigenvalues and eigenvectors

Tensor field

Curvature

Weyl tensor

Geometry

Tensor product

Exact solutions in general relativity

Quantum mechanics

Cartesian tensor

Linear and Multilinear Algebra

Recently, the inverse of an even-order square tensor has been put forward in [Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl. 2013;34(2):542–570] by means of the tensor group consisting of even-order square tensors equipped with the Einstein product. In this paper, several necessary and sufficient conditions for the invertibility of a tensor are obtained, and some approaches for calculating the inverse (if it exists) are proposed. Furthermore, the Cramer's rule and the elimination method for solving the tensor equations with the Einstein product are derived. In addition, the tensor eigenvalue problem mentioned in [Qi L-Q. Theory of tensors (hypermatrices). Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University; 2014] can also be addressed by using the elimination method mentioned above. By the way, the LU decomposition and the Schur decomposition of matrices are extended to tensor case. Numerical examples are provided to illustrate the main results.

Tensor inversion and its application to the tensor equations with Einstein product