Eigenvalues And Singular Values Of Tensors

Tensor analysis is an important tool in theoretical physics, continuum mechanicsand many other areas of science and engineering. The study of eigenvalues and singu-lar values of high-order tensors has become a significant topic in applied mathematicsand numerical multilinear algebra. In recent years, rectangular tensors arise from thestrong ellipticity condition problem in solid mechanics and the entanglement problemin quantum physics. In this paper, we mainly investigate the eigenvalues of squaretensors, diagonally dominant square tensors and nonnegative rectangular tensors andE singular values of rectangular tensors. The full text is divided into four parts.In the first part, we briefly introduce the research background and current situationof the tensor, and probe into the main content of the paper.In the second part, we introduce the basic concepts of tensors.In the third part, we consider the square tensors. At first, we introduce the bound-s for the eigenvalues of a square tensor, and thus deduce some conclusions about adiagonally dominant tensor.In the fourth part, we mainly consider the rectangular tensors. we study thespectral radius of a nonnegative rectangular tensor and E singular values of a realpartially symmetric rectangular tensor respectively.