On Scaled Almost-Diagonal Hermitian Matrix Pairs

This paper contains estimates concerning the block structure of Hermitian matrices H and M, which make a scaled diagonally dominant definite pair. The obtained bounds are expressed in terms of relative gaps in the spectrum of the pair (H,M) and norms of certain blocks of the matrices DHD and DMD, where D is either $[|\mbox{diag(H)}|]^{-{1}/{2}}$ or $[\mbox{diag(M)}]^{-{1}/{2}}$. If either of the matrices H, M is diagonal, the new results assume simple and applicable form. For scaled diagonally dominant Hermitian matrices, the new estimates compare favorably with the existing ones for accurate location of the smallest eigenvalues.