On Some Results Related to Positive Definite Matrices by Means of Cauchy-Schwarz Inequality

Abstract

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 semi-definite matrices are investigated.