Geodesic
A generalization of Weyl's inequalities with implications
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIII, Tome 248 (1998), pp. 49-59

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This paper suggests a generalization of additive Weyl's inequalities to the case of two square matrices of different orders. As a consequence of generalized Weyl's inequalities, a theorem describing the location of eigenvalues of a Hermitian matrix in terms of the eigenvalues of an arbitrary Hermitian matrix of smaller order is derived. It is demonstrated that the latter theorem provides a generalization of Kahan's theorem on clustered eigenvalues. Also it is shown that the theorem on extended interlacing intervals established in [3] is another consequence of the generalized additive Weyl inequalities suggested. 
@article{ZNSL_1998_248_a2,
     author = {L. Yu. Kolotilina},
     title = {A generalization of {Weyl's} inequalities with implications},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {49--59},
     publisher = {mathdoc},
     volume = {248},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1998_248_a2/}
}
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L. Yu. Kolotilina. A generalization of Weyl's inequalities with implications. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIII, Tome 248 (1998), pp. 49-59. http://geodesic.mathdoc.fr/item/ZNSL_1998_248_a2/