Majorization and Singular Values

A citation from von Neumann: “The object of this note is the study of certain properties of complex matrices of $$n$$ th order: $$A=(a_{ij})_{i,j=1}^n$$ , $$n$$ being a finite positive integer: $$n=1,2,\dots $$ . Together with them we shall use complex vectors of $$n$$ th order (in $$n$$ dimensions): $$x=(x_i)_{i=1}^n$$ .” This classical subject in matrix theory is exposed in Sects. 6.2 and 6.3 after discussions on vectors in Sect. 6.1. This chapter also contains several matrix norm inequalities as well as majorization results for matrices, which were mostly developed more recently.