Matrix Singular Value And Eigenvalue Estimates,

This paper investigates some properties about the singular value and eigenvalue of matrices.Firstly, generalize Shu-Lin Liu's result and improve his result, obtain the greatest eigenvalue of a nonnegative matrix. It can be appliable to every nonnegative matrix, the new bounds are sharper and the proof is more simple than the Frobenius theorem.Secondly, construct an interval in such a way that all the singular values are contained in each interval, and this interval is a decreasing sequence.Then, derive monotonic sequence of bounds for the biggest,the smallest eigenvalues of n×n nonsingular complex matrix and estimate the inclusion regions of eigenvalues.Finally, bounds forσ_k~2 ( A) ...σ_l~2( A) andσ_k~2 ( A) +...+σ_l~2( A), involving k , l , n , A F and det A only, are obtained. And when k = l= 1, as special cases, we can obtain bounds for individual singular values.