Brualdi-type Eigenvalue Inclusion Sets Of Tensors

The concept of tensors was introduced by Gauss, Riemann and Christoffel, etc., in the 19th century in the study of differential geometry. In the very beginning of the 20th century,Ricci and Levi-Civita,ect., further developed tensor analysis as a mathematical discipline. It was Einstein who applied tensor in his study of general relativity in 1913. This made tensor analysis an important tool in theoretical physics, continuum mechanics and many other areas of science and engineering.At the beginning of the century, the calculation of tensor(also called numerical multilinear algebra) have became a new branch in applied mathematics and computational mathematics.Because of the need of large data analysis, Golub Gene of Stanford University in the United States lead a group of researchers who come from Europe and United States, beginning of this research, the main topic are the tensor decomposition and tensor approximation.In 2005, Liqun Qi and Lim defined the eigenvalue of tensors, respectively, and some properties for eigenvalues of tensor are discussed. Tensor is a multi-way array and can be regard as the extension of matrix. The classical results in the matrix are also set up for tensor,which is worthy of study. In 2005, Liqun Qi gave Gersgorin-type eigenvalue inclusion sets of tensors. Since 2008, Zhang Gongqing, etc., prove that the famous Frobenius-Perron theorem of the nonnegative matrix theory hold for tensors, giving a series of results. As a frontier research subject, the eigenvalues of tensor are faced many problems, which need to be solved by new methods and new technologies.The innovation of this paper, we first use graph of tensor and combinatorial methods to study the eigenvalue of tensor, prove that the famous Brualdi type eigenvalue inclusion set of nonnegative matrix hold for tensor and give the Brualdi type inclusion set of tensor and other relevant conclusions. This paper consists of three chapters.In the first chapter, we introduces the research background, significance and the current research situation of the study.In the second chapter, we introduces some theorem of matrix and the definition of the tensor, basic knowledge, which lays the groundwork for the third Chapter.In the final chapter, we give the Brualidi type eigenvalue inclusion sets of tensors, by using the digraphs and combination method of the tensor.