Eigenvalue Inequalities and Schubert Calculus

Using techniques from algebraic topology we derive linear inequalities which relate the spectrum of a set of Hermitian matrices A₁,…, A_r ? ¢^n×n with the spectrum of the sum A₁ + … + A_r. These extend eigenvalue inequalities due to Freede-Thompson and Horn for sums of eigenvalues of two Hermitian matrices.