The Extremely Eigenvalue Estimates And Inverse N Matrix <sub> 0 </ Sub> <sup> - </ Sup> Matrix Perron I Study

This paper investigates the bounds for the extreme eigenvalues and the Perron complement of inverse N₀-matrices, including the bounds for the Perron root of nonnegative matrix and the minimal eigenvalue and some related results involving the Perron complement of inverse N₀-matrices. The main contents are as follows:1. The background of this paper and some current state of eigenvalue and relevant research performance are summarized, and our purposes of this paper are presented.2. For the Perron root of nonnegative irreducible matrices, three sequences of lower bounds are presented by means of constructing shifted matrices, which convergence is studied. The comparisons of the sequences with known ones are supplemented with a numerical example.3. Simple estimates of lower bounds for the minimal eigenvalue are presented based on the arithmetic-geometric-mean inequality. The results, involving the trace and determinant, are improved by the method which is used to estimate the smallest singular value. An example shows that this method is valid.4. Two inequalities for the minimal eigenvalue of M-matrices are given. Based on the previous researches, we obtain an increasing sequence of lower bounds for the minimal eigenvalue of M-matrices.5. The Perron complement of inverse N₀-matrices are considered and some related inequalities involving N₀-matrices and inverse N₀-matrices are given.