Singularity, Wielandt’s lemma and singular values

Abstract

In this study, some upper and lower bounds for singular values of a general complex matrix are investigated, according to singularity and Wielandt’s lemma of matrices. Especially, some relationships between the singular values of the matrix A and its block norm matrix are established. Based on these relationships, one may obtain the effective estimates for the singular values of large matrices by using the lower dimension norm matrices. In addition, a small error in Piazza (2002) [G. Piazza, T. Politi, An upper bound for the condition number of a matrix in spectral norm, J. Comput. Appl. Math. 143 (1) (2002) 141–144] is also corrected. Some numerical experiments on saddle point problems show that these results are simple and sharp under suitable conditions.