Localization And Estimating Of Eigenvalues Of Matrices And Higher Order Tensors

Matrix eigenvalue problem, which contains mainly two aspects:Localization of matrix eigenvalues, that is, giving a region which contains all eigenvalues of a matrix in the complex plane; Computing (Estimating) of matrix eigenvalues, that is, giving an approximation (a bound) for an eigenvalue of a matrix, is not only an important branch of Matrix theory, but also have applications in many fields. In this dissertation, we localize and estimate the eigenvalues of matrices and obtain the following results: first, two eigenvalue inclusion regions for matrices are given, and it is proved that these regions are tighter than those of the well known Gersgorin eigenvalue inclusion theorem, Brauer eigenvalue inclusion theorem, and the theorems in [L. Cvetkovic, V. Kostic, R. Bru and F. Pedroche. A simple generalization of Gersgorin's theorem. Adv. Comput. Math.,2011,(35):271-280]. Furthermore, by the technique of partitioning matrices, the other two eigenvalue inclusion regions are given, and it is proved that these regions are tighter than that of Gersgorin eigenvalue inclusion theorem for partitioned matrices. Second, some bounds for eigenvalues of matrices having Perron—Frobenius property and Generalized M-matrices are given, and it is proved that these bounds are sharper than those in [D. Noutsos. On Perron—Frobenius property of matrices having some negative entries. Linear Algebra Appl.,2006,(412):132-15] and [G.X. Tian and T.Z. Huang. Inequalities for the minimum eigenvalue of M-matrices. Electron. J. Linear Algebra,2010,(20):291-302], respectively.The class of higher order tensors as a generalization of matrices, has a wide range of practical applications, such as signal processing, data analysis and data mining. In this dissertation, we study the localization and estimating of tensor eigenvalues, and obtain two tensor eigenvalue inclusion theorems which provide two inequalities to identify the positive definiteness of an even-degree homoge-neous polynomial form. Meanwhile, we construct an algorithm to identify the positive definiteness of an even-degree homogeneous polynomial form. Numerical examples are given to verify its feasibility and efficiency. Furthermore, new up-per and lower bounds for a nonnegative tensor are given, and it is proved that these bounds are sharper than those of [Y. Yang and Q. Yang. Further results for Perron—Frobenius theorem for nonnegative tensors. SIAM. J. Matrix Anal. Appl.,2010,(31):2517-2530].