Augmentation and Modification Problems for Hermitian Matrices

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order $r$ for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian $((n+r)\times (n+r))$-matrix with a given $(n\times n)$-block $A_{11}$ in block $(2\times 2)$-representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian $(n\times n)$-matrix $B$ of rank not greater than $r$ so that the obtained matrix, being added to a given Hermitian $(n\times n)$-matrix $A$, will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.