Estimation Of Eigenvalue Bounds For Several Types Of Special Matrices

Nonnegative matrices and M-matrices are special matrices with important application background.Many problems in biology,physics and economy science and so on have close connection with nonnegative matrix and M-matrix.The Hadamard product and Fan product of M-matrix and nonnegative matrices are important problems in matrix theory.Especially,the smallest eigenvalue(?(Bo A^-1))problem of the Hadamard product of M-matrix and the inverse of M-matrix,the minimum eigenvalue(?(Ao A^-1))problem of the Hadamard product of M-matrix and its inverse,the minimum eigenvalue???A?B??problem of the Fan product of M-matrix,the minimum eigenvalue???B??problem of M-matrix,as well as the spectral radius???Ao B??problem of the Hadamard product of nonnegative matrices have been extensivelly concerned and researched in recent years,and some important results are obtained.In this thesis,we continue to research these problems,and some more accurate estimate formulas of?(Ao A^-1),?(Bo A^-1),??A?B?,??B?and??Ao B?are presented which are easily to calculate.Meanwhile,we also carry on discussion and numerical verification to these estimation formulas.Our main results are as follows.Firstly,for the Hadamard product Ao Bof nonnegative matrices Aand B,some new upper bounds for the spectral radius of Ao B are obtained by using the theorem for inclusion region of a matrix.Numerical examples show that these estimating formulas improve the related results in some cases.Secondly,for the Hadamard productBo A^-1of M-matrices A and B,some new upper bounds for the the minimum eigenvalue ofBo A^-1andAo A^-1are obtained by using the theorem for inclusion region of a matrix.Theoretical proof and numerical examples show that these estimating formulas improve the related results in some cases.Thirdly,for the Fan product A?B of M-matrices A and B,some new upper bounds for the the minimum eigenvalue of A?B are obtained by using the theorem for inclusion region of a matrix.Numerical examples show that our derived results improve some existing ones in some cases.Lastly,for the Hadamard productA?B^-1of a nonnegative matrix A and an M-matrix B,some new upper bounds for the spectral radius ofA?B^-1are obtained by using optimally scaled matrix,Jacobi iterative matrix and the relationship between matrix eigenvalue and eigenvector.Meanwhile,some new lower bounds for the minimum eigenvalue ofan M-matrix B are given.Numerical examples show that these estimating formulas improve the related results in some cases,and the bounds only depend on the entries of matrices,therefore,they are easy to calculate.