Matrices with some extremal properties

AbstractFor a complex matrix A, there are many inequalities related to its eigenvalues, diagonal elements, and three types of singular values. We investigate the conditions on the matrices for which those inequalities become equalities. It is shown that in many cases the conditions are both necessary and sufficient for the matrices to be unitarily similar to matrices of lower orders. Moreover, we get alternative proofs of various characterization theorems of different classes of matrices