Eigenvalue Inequalities of Matrix Product

Eigenvalue problem of  matrices  is of central importance in matrix theory, matrix computation, linear algebra and related areas. Traditionally,eigenvalue inequalities in partial sums involve  two Hermitian matrices for sum and a pair of positive semidefinite matrices  for product. We show some eigenvalue inequalities of the product of a Hermitian matrix and a positive definite (Hermitian) matrix; we use the results to study perturbation problems of generalized eigenvalues. （Joint work with Bo-Yan Xi.）