Eigenvalue bounds and inequalities using vector aggregation of matrices

For matrices partitioned into block form, the operation of (block)-vector aggregation, which associates with a given matrix a matrix of smaller size, is introduced. Properties of aggregated matrices are analyzed. In particular, it is shown that in the Hermitian case, the eigenvalues of a block-vector-aggregated matrix interlace those of the original matrix. By using vector aggregation, new eigenvalue bounds and inequalities for normal and Hermitian matrices are derived and put in context with the existing ones. In particular, inequalities interrelating eigenvalues of a block-partitioned Hermitian matrix with those of its diagonal blocks are obtained. Also it is shown that the spectral constants characterizing the block partitioning of a Hermitian matrix are bounded below by the corresponding constants related to associated vector-aggregated matrices.