Eigenvalue Problems Of Nonnegative 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, Levi-Civita, etc., further developed tensor analysis as a mathematical discipline. It was Einstein who applied tensor analysis in his study of general relativity in 1916. This made tensor analysis an important tool in theoretical physics, continuum mechanics and many other areas of science and engineering [5, 7,9,19].Eigenvalues of higher order tensors have become an important topic of study in a new applied mathematics branch, numerical multilinear algebra, and they have a wide range of practical applications. In recent studies of numerical multilinear al-gebra, eigenvalue problems for tensors have attracted special attention. Qi [15] gave the definition of symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor and discussed the properties of eigenvalues. He defined the rank of tensor in [14,16]. Chang et al. discussed the problems of multiplicity of eigenvalues for tensors [2,3]. In [2], the Perron-Frobenius theorem for nonneg-ative matrices has been generalized to the class of nonnegative tensors. The same conclusion was also discussed in [10].In this thesis, we discuss some eigenvalue problems of nonnegative tensors. In chapter 1, we introduce the definition of tensors. Its applications and study area are shown here. In chapter 2, we first review the Perron-Frobenius theorem for nonnega-tive tensors, then we give further results on it. Next we generalize the Ky Fan theorem from nonnegative matrices to nonnegative tensors. Further results on maxmin prob-lems are discussed. Last we give an equivalent condition for nonnegative irreducible tensors. The definition of spectral radius of tensors is given here. Totally speaking, these results enhance the theory of nonnegative tensors. In chapter 3, we give an algorithm to find out the spectral radius of a class of nonnegative tensors. The proof shows that our algorithm can assure to find out the spectral radius and its correspond-ing eigenvector for positive tensors. Numerical results are presented in the last.