On some results related to positive semidefinite matrices by means of Cauchy-Schwarz inequality

In this study, the relationship between the Cauchy-Schwarz inequality and matrix norms and singular values is discussed. First, the classical form of the Cauchy-Schwarz inequality is recalled, and its extended versions in matrix space are examined. Then, the connections between singular values and unitarily invariant norms are addressed, and various inequalities are derived in this context. In particular, the majorization properties of singular values for Hermitian and positive semidefinite matrices are investigated.