Chan, Ka Kwan Kevin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 84-86). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Preliminaries --- p.6 / Chapter 2.1 --- Eigenvalues of Sums of Hermitian Matrices --- p.6 / Chapter 2.2 --- Highest Weights --- p.12 / Chapter 2.3 --- Schubert Calculus --- p.15 / Chapter 2.4 --- Invariant Factors --- p.18 / Chapter 2.5 --- Singular Values of Sums and Products --- p.19 / Chapter 2.6 --- Relations Among the Problems --- p.25 / Chapter 3 --- Klyachko's Results --- p.27 / Chapter 3.1 --- Klyacho's Proof --- p.27 / Chapter 3.1.1 --- Rayleigh Trace --- p.28 / Chapter 3.1.2 --- Facts from Vector Bundles and Geometric Invariant Theory --- p.31 / Chapter 3.1.3 --- Proof of Klyachko's Results --- p.33 / Chapter 3.2 --- Proof by Symplectic Geometry --- p.42 / Chapter 4 --- Saturation Conjecture --- p.47 / Chapter 4.1 --- Definitions --- p.48 / Chapter 4.2 --- Proof of the Conjecture --- p.52 / Chapter 4.3 --- Remarks --- p.58 / Chapter 5 --- Proof of the Theorems --- p.60 / Chapter 5.1 --- Main Results --- p.60 / Chapter 5.2 --- "Real Symmetric and Quaternionic Hermitian Matrices, Compact Operators" --- p.70 / Chapter 5.3 --- Highest Weights --- p.71 / Chapter 5.4 --- Schubert Calculus --- p.71 / Chapter 5.5 --- Invariant Factors --- p.72 / Chapter 5.6 --- Singular Values of Sums and Products --- p.75 / Chapter 6 --- Further Problems --- p.79 / Bibliography --- p.84