Generalized Wielandt and Cauchy-Schwarz inequalities

The Wielandt inequality is important in many applications. It involves functions of the extreme eigenvalues of a positive definite matrix. In this paper, we derive a few extensions of the Wielandt inequality and new inequalities involving the two largest and two smallest eigenvalues. The resulting inequalities are shown to be the best possible. A unified approach involving constrained optimization techniques are used to derive these results. The proposed inequalities are then utilized to obtain several bounds for the extremum eigenvalues and eigen spread of real symmetric matrices. A collection of bounds for functions of the eigenvalues of positive definite and general symmetric matrices are then derived in terms of the entries of the matrix. Additionally, lower bounds for the condition number of positive definite matrices as well as lower bounds for the minimum separation of eigenvalues are developed.