Geodesic
Augmentation and Modification Problems for Hermitian Matrices
Matematičeskie zametki, Tome 71 (2002) no. 1, pp. 130-134.

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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. 
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     title = {Augmentation and {Modification} {Problems} for {Hermitian} {Matrices}},
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E. E. Tyrtyshnikov; V. N. Chugunov. Augmentation and Modification Problems for Hermitian Matrices. Matematičeskie zametki, Tome 71 (2002) no. 1, pp. 130-134. http://geodesic.mathdoc.fr/item/MZM_2002_71_1_a11/

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