Abstract
The pioneering work of Issai Schur (1923) on majorization was 1 motivated by his discovery that the eigenvalues of a positive semidefi- 2 nite Hermitian matrix majorize the diagonal elements. This discovery 3 provided a new and fundamental understanding of Hadamard’s deter- 4 minant inequality that led Schur to a remarkable variety of related 5 inequalities. Since Schur’s discovery, a number of other majorizations 6 have been found in the context of matrix theory. These majorizations 7 primarily involve quantities such as the eigenvalues or singular val- 8 ues of matrix sums or products. An integral part of the development 9 of majorization in matrix theory is the extremal representations of 10 Chapter 20.