Eigenvalues of sum of hermitian matrices

Abstract

The research deals with a systematic provision of mathematical algorithms to solve eigenvalue inequalities bounding the sum of eigenvalues of Hermitian matrices. We study the problem of defining the set of eigenvalues of sum of Hermitian matrices and then investigate the relationship existing between the eigenvalues of ni × nj Hermitian matrices A and B to the eigenvalues of their summand H = A+B. The recent solution appears to deal with the situation of eigenvalue inequalities where r = n − 2. We modified a number of numeri cal methods to formulate inequalities connecting sum of r eigenvalues for H in relation to sum of r eigenvalues for both A and B depending on the parameters for r < n where r = n−1,n−3. Lastly, the research showed the solution of possible eigenvalues of sum of Hermitian matrices of size Hn r for n ≤ 12. Other gaping issues were also specified.