Spectral Radius Estimation Of Non-negative Matrices And Tensors

The nonnegative matrix is a kind of important matrix among matrices.And estimating its spectral radius is a main study object in nonnegative matrix theory.There have been many research literatures to estimate spectral radius of the non-negative matrix recently.Tensor,as the extension of Inatrix,its spectral radius can be utilized widely in science and engineering projects and has become an attractive research subject.In order to study the spectral radius of natrix and tensor,the specific work of this paper are as follows:Chapter one,some basic definitions and characters of nonnegative matrix and tensor are introduced,such as,the spectral radius of matrix,the H-eigenvalue of tensor,the Z-eigenvalue of tensor,a product of tensor and some relevant important theorems.Chapter two,the new estimation of ?(A)of nonnegative irreducible matrix is studied.At first,two new matrices B=(A2+A-al)n-1 and C=(A+I)n-1 are constructed,where ? is the sum of the minimum elements of matrices A and A2.According to Perron-Frobenius theorem and the related lemma of nonnegative matrix,the new upper and lower bound of ?(A)of nonnegative irreducible matrices are obtained.And the astringency of the new upper and lower bound of the new estimation formula are pointed out,Moreover,the estimation of ?(A)is more accu-rate as the parameters s and k are larger.At last,by using numerical examples,the estimation method is more accurate than the results of existing relevant.literatures to some extent.Chapter three,the spectral radius of nonnegative tonsor is estimated.At first,On the basis of the Perron-Frobenius theorem of nonnegative weakly symmetric irreducible tensor Z-eigenvalue and the relation of the product of tensor and Z-eigenvalue,the upper bound of the Z-spectral radius of nonnegative weakly sym-metric irreducible tensor is given.Next,according to the character of diagonal similarity tensor mid the relation of its and H-spectral radius,the esti-mtion of nonnegative weakly irreducible tensor H-spectral radius is gained.At last,numerical examples are given to demonstrate the effectiveness of the proposed estimation method.