Non-negative Matrix Spectral Radius Is Estimated

Matrix is an important concept in modern mathematics, the theories and methods of which deal with so many complicated problem in real applications. Matrix with merits of concise expressions and comprehensive descriptions, etc., has become a very important mathematical tool in coping with mathematical problems and in the field of engineering and technology.Theory of matrix is a vital mathematical theory. Plenty of practical problems often reduce to one or a few large-scale systems of matrix (nonnegative matrix) at last Nonnegative matrix is a crucial class of matrix in theory of matrix. Research of nonnegative matrix has been continued and the research of the solutions of the eigenvalues of nonnegative matrix and their estimates is momentous subject in the research fields of Matrix analysis and Numerical algebra. Meanwhile, the estimates of Perron root and research of Perron complement of nonnegative matrix have been thought more and more by a few practitioners. Based on the wide applications of the estimates of the spectrum of nonnegative matrix in so many fields, especially the estimates of the upper and lower bounds of Perron root.This paper mainly has an investigation in the estimates of Perron root and Perron complement of nonnegative matrix, obtaining a few better conclusions. Discussion of the estimate expressions of the spectrum radius of block nonnegative matrix, the guess conclusions of which improve the classical Frobenius results, followed by a series of monotone increasing lower bounds and a series of monotone decreasing upper bounds on nonnegative matrix. Improvement of the results obtained by Fujian Duan, Kecun Zhang by means of similar diagonalization transformation and Gerschgorin Theory with a parameter introduced, obtaining a series of upper bounds of the spectrum radius of nonnegative matrix. Based on the property of the characteristic root and the corresponding characteristic vector, new bounds of the spectral radius are obtained.